Fundamentals of Economic Model Predictive Control - Semantic Scholar
Fundamentals of Economic Model Predictive Control James B. Rawlings, David Angeli and Cuyler N. Bates Dept. of Chemical and Biological Engineering, Univ. of Wisconsin-Madison, WI, USA
Dept. of Electrical and Electronic Engineering, Imperial College London, UK
CDC Meeting Maui, HI December 10-14, 2012 Rawlings/Angeli/Bates
Economic MPC
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Outline 1
Introduction to MPC and economics
2
Stability of standard (tracking) MPC
3
Unreachable setpoints and turnpikes
4
Economic MPC
5
Dissipativity
6
Average constraints
7
Periodic terminal constraint
8
Conclusions and open research issues
Rawlings/Angeli/Bates
Economic MPC
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Optimizing economics: current industrial practice
Planning and Scheduling 1
Steady State Optimization
Model Update
Validation
Reconciliation
Two layer structure I
Steady-state layer F
F
RTO optimizes steady state model Optimal setpoints passed to dynamic layer
Controller
Plant
Rawlings/Angeli/Bates
Economic MPC
3 / 94
Optimizing economics: current industrial practice
Planning and Scheduling 1
Steady State Optimization
Model Update
Validation
Reconciliation
Two layer structure I
Steady-state layer F
F
I
Controller
Dynamic layer F
F
Plant
Rawlings/Angeli/Bates
RTO optimizes steady state model Optimal setpoints passed to dynamic layer
Economic MPC
Controller tracks the setpoints Linear MPC
3 / 94
Optimizing economics: current industrial practice
Planning and Scheduling 1
Steady State Optimization
Model Update
Validation
Reconciliation
2
Two layer structure Drawbacks
Controller
Plant
Rawlings/Angeli/Bates
Economic MPC
4 / 94
Optimizing economics: current industrial practice
Planning and Scheduling 1
Steady State Optimization
Model Update
Validation
Reconciliation
2
Two layer structure Drawbacks I I
I
Controller I
Inconsistent models Re-identify linear model as setpoint changes Time scale separation may not hold Economics unavailable in dynamic layer
Plant
Rawlings/Angeli/Bates
Economic MPC
4 / 94
Steady-state optimization problem definition
Stage cost: `(x, u) Optimization: (xs , us ) = arg min x,u
`(x, u)
subject to: x = f (x, u),
Rawlings/Angeli/Bates
Economic MPC
(x, u) ∈ Z
5 / 94
Tracking MPC
Setpoint Outputs Inputs
← Past
y u
k=0
Future →
One step of a closed-loop MPC trajectory
Rawlings/Angeli/Bates
Economic MPC
6 / 94
Tracking MPC problem definition Stage cost: `t (x, s) = |x(k) − xs |2Q + |u(k) − us |2R + |u(k) − u(k − 1)|2S
Rawlings/Angeli/Bates
Economic MPC
7 / 94
Tracking MPC problem definition Stage cost: `t (x, s) = |x(k) − xs |2Q + |u(k) − us |2R + |u(k) − u(k − 1)|2S Optimization: min VN (x, u) = u
N−1 X
`t (x(k), u(k))
k=0
+ x = f (x, u) (x(k), u(k)) ∈ Z k ∈ I0:N−1 subject to x(N) = xs x(0) = x Control law: u = κN (x) Admissible set: XN Rawlings/Angeli/Bates
Economic MPC
7 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller!
Rawlings/Angeli/Bates
Economic MPC
8 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
Rawlings/Angeli/Bates
Economic MPC
8 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
x2
k 0
x1
Rawlings/Angeli/Bates
Economic MPC
8 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
x2
k +1 k 0
x1
Rawlings/Angeli/Bates
Economic MPC
8 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
x2
k +2 k +1 k 0
x1
Rawlings/Angeli/Bates
Economic MPC
8 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
x2
closed-loop trajectory k +2 k +1 k 0
x1
Rawlings/Angeli/Bates
Economic MPC
8 / 94
Infinite horizon solution The infinite horizon ensures stability
Rawlings/Angeli/Bates
Economic MPC
9 / 94
Infinite horizon solution The infinite horizon ensures stability Open-loop predictions equal to closed-loop behavior
Rawlings/Angeli/Bates
Economic MPC
9 / 94
Infinite horizon solution The infinite horizon ensures stability Open-loop predictions equal to closed-loop behavior May be difficult to implement
x2
0 k
Φk x1 Rawlings/Angeli/Bates
Economic MPC
9 / 94
Infinite horizon solution The infinite horizon ensures stability Open-loop predictions equal to closed-loop behavior May be difficult to implement
x2
0
k +1 k
Vk+1 = Vk − L(xk , uk ) Φk x1 Rawlings/Angeli/Bates
Economic MPC
9 / 94
Infinite horizon solution The infinite horizon ensures stability Open-loop predictions equal to closed-loop behavior May be difficult to implement
x2 Vk+2 = Vk+1 − L(xk+1, uk+1) k +2 0
k +1 k
Vk+1 = Vk − L(xk , uk ) Φk x1 Rawlings/Angeli/Bates
Economic MPC
9 / 94
Terminal constraint solution Adding a terminal constraint ensures stability
Rawlings/Angeli/Bates
Economic MPC
10 / 94
Terminal constraint solution Adding a terminal constraint ensures stability May cause infeasibility
Rawlings/Angeli/Bates
Economic MPC
10 / 94
Terminal constraint solution Adding a terminal constraint ensures stability May cause infeasibility Open-loop predictions not equal to closed-loop behavior
x2
k 0
Φk x1 Rawlings/Angeli/Bates
Economic MPC
10 / 94
Terminal constraint solution Adding a terminal constraint ensures stability May cause infeasibility Open-loop predictions not equal to closed-loop behavior
x2
k +1 k 0 Vk+1 ≤ Vk − L(xk , uk ) Φk x1 Rawlings/Angeli/Bates
Economic MPC
10 / 94
Terminal constraint solution Adding a terminal constraint ensures stability May cause infeasibility Open-loop predictions not equal to closed-loop behavior
x2 Vk+2 ≤ Vk+1 − L(xk+1, uk+1) k +2 k +1 k 0 Vk+1 ≤ Vk − L(xk , uk ) Φk x1 Rawlings/Angeli/Bates
Economic MPC
10 / 94
Closed-loop stability of tracking MPC Assumption: Model, cost and admissible set 1
2
The model f (·) and stage cost `(·) are continuous. The admissible set XN contains xs in its interior. There exists a set Xf containing xs in its interior and K∞ -function γ(·) such that VN0 (x) ≤ γ(|x − xs |) for x ∈ Xf .
Rawlings/Angeli/Bates
Economic MPC
11 / 94
Closed-loop stability of tracking MPC Assumption: Model, cost and admissible set 1
2
The model f (·) and stage cost `(·) are continuous. The admissible set XN contains xs in its interior. There exists a set Xf containing xs in its interior and K∞ -function γ(·) such that VN0 (x) ≤ γ(|x − xs |) for x ∈ Xf .
Theorem: Stability of tracking MPC with terminal constraint The steady-state target (xs , us ) is an asymptotically stable equilibrium point of the closed-loop system x + = f (x, κN (x)) with region of attraction XN . Rawlings/Angeli/Bates
Economic MPC
11 / 94
Setpoints and unreachable setpoints
Rawlings/Angeli/Bates
Economic MPC
12 / 94
Setpoints and unreachable setpoints
Consider the steady state of a linear dynamic model with state x, controlled input u, and disturbance w x(k + 1) = Ax(k) + Bu(k) + Bd w (k)
Rawlings/Angeli/Bates
Economic MPC
12 / 94
Setpoints and unreachable setpoints
Consider the steady state of a linear dynamic model with state x, controlled input u, and disturbance w x(k + 1) = Ax(k) + Bu(k) + Bd w (k) xs = (I − A)−1 B us + (I − A)−1 Bd ws | {z } | {z } G
ds
xs = Gus + ds
Rawlings/Angeli/Bates
Economic MPC
12 / 94
Steady states—unconstrained system xs xs = Gus + ds ds1 = 1
ds2 = 0
ds3 = −1
G xsp
us1
us2
us3 us
For an unconstrained system with G 6= 0, any setpoint xsp with any disturbance ds has a corresponding us . Rawlings/Angeli/Bates
Economic MPC
13 / 94
Constraints and unreachable setpoints xs xs = Gus + ds ds1 = 1
0 ≤ us ≤ 1
ds2 = 0
ds3 = −1
G xsp
0
us1
1 us2 us3
us
For a constrained system, the setpoint xsp may be unreachable for a given disturbance ds . MPC is method of choice for this situation. Rawlings/Angeli/Bates
Economic MPC
14 / 94
Constraints and unreachable setpoints xs xs = Gus + ds 0 ≤ us ≤ 1 ds ≥ G
0 ≤ ds ≤ G
xsp
0
1 us ds ≤ 0
As the estimated disturbance changes with time, the setpoint may change between reachable and unreachable. Rawlings/Angeli/Bates
Economic MPC
15 / 94
What closed-loop behavior is desirable? Fast tracking
xsp x(0) Q R (fast tracking) x∗
x
x(0)
k
Rawlings/Angeli/Bates
Economic MPC
16 / 94
What closed-loop behavior is desirable? Slow tracking
xsp x(0) Q R (slow tracking) x∗
x
x(0)
k
Rawlings/Angeli/Bates
Economic MPC
17 / 94
What closed-loop behavior is desirable? Asymmetric tracking
xsp x(0) Q R (fast tracking) x∗
x
x(0)
k
Rawlings/Angeli/Bates
Economic MPC
18 / 94
Creating a turnpike example
Standard linear quadratic problem x + = Ax + Bu `(x, u) = |Cx − ysp |2Q + |u − usp |2R
Rawlings/Angeli/Bates
Economic MPC
Q > 0, R > 0
19 / 94
Creating a turnpike example
Standard linear quadratic problem x + = Ax + Bu `(x, u) = |Cx − ysp |2Q + |u − usp |2R
Q > 0, R > 0
Choose an inconsistent setpoint A = 1/2
B = 1/4 ys = Gus usp = 0
Rawlings/Angeli/Bates
C =1
Q=1
R=1
G = 1/2 ysp = 2
Economic MPC
19 / 94
Inconsistent setpoint and optimal steady state (usp, xsp)
Optimal steady state
x
usp = 0 us = 0.8
xsp = 2 xs = 0.4
(us , xs ) G u
Rawlings/Angeli/Bates
Economic MPC
20 / 94
Optimal control problem
Cost function and dynamic model VN (x, u) =
N−1 X
`(x(k), u(k))
s.t. x + = Ax + Bu,
x(0) = x
k=0
Rawlings/Angeli/Bates
Economic MPC
21 / 94
Optimal control problem
Cost function and dynamic model VN (x, u) =
N−1 X
`(x(k), u(k))
s.t. x + = Ax + Bu,
x(0) = x
k=0
Optimal state and input trajectories min V (x, u) u
Rawlings/Angeli/Bates
u0 (x),
x0 (x)
Economic MPC
21 / 94
Optimal trajectory: xsp = 2, usp = 0 1 0.5
x
0
N =5
-0.5 -1 0
u
1
1 0.9 0.8 0.7 0.6 0.5 0.4
2
3
4
2 t
3
4
x0 = −1 x0 = 1
0 Rawlings/Angeli/Bates
1
Economic MPC
22 / 94
Optimal trajectory: xsp = 2, usp = 0 1 0.5
x
N = 30
0 -0.5 -1 0
u
1 0.9 0.8 0.7 0.6 0.5 0.4
5
10
15
20
25
30
10
15 t
20
25
30
x0 = −1 x0 = 1
0 Rawlings/Angeli/Bates
5
Economic MPC
22 / 94
Optimal trajectory: xsp = 2, usp = 0 1 0.5
x
N = 100
0 -0.5 -1 0
u
20
30
40
50
60
70
80
90 100
1 0.9 x0 = −1 0.8 0.7 x0 = 1 0.6 0.5 0.4 0 10 20
30
40
50 t
60
70
80
90 100
Rawlings/Angeli/Bates
10
Economic MPC
22 / 94
Introduction to Turnpike Literature
It is exactly like a turnpike paralleled by a network of minor roads.
Rawlings/Angeli/Bates
Economic MPC
23 / 94
Introduction to Turnpike Literature
It is exactly like a turnpike paralleled by a network of minor roads. There is a fastest route between any two points; and if the origin and destination are close together and far from the turnpike, the best route may not touch the turnpike.
Rawlings/Angeli/Bates
Economic MPC
23 / 94
Introduction to Turnpike Literature
It is exactly like a turnpike paralleled by a network of minor roads. There is a fastest route between any two points; and if the origin and destination are close together and far from the turnpike, the best route may not touch the turnpike. But if the origin and destination are far enough apart, it will always pay to get on the turnpike and cover distance at the best rate of travel, even if this means adding a little mileage at either end.
Rawlings/Angeli/Bates
Economic MPC
23 / 94
Introduction to Turnpike Literature
It is exactly like a turnpike paralleled by a network of minor roads. There is a fastest route between any two points; and if the origin and destination are close together and far from the turnpike, the best route may not touch the turnpike. But if the origin and destination are far enough apart, it will always pay to get on the turnpike and cover distance at the best rate of travel, even if this means adding a little mileage at either end. —Dorfman, Samuelson, and Solow (1958, p.331)
Rawlings/Angeli/Bates
Economic MPC
23 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply Simulations indicate the closed loop is stable
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply Simulations indicate the closed loop is stable How can we be sure?
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—stability result (linear model)
Theorem: Asymptotic Stability of Terminal Constraint MPC The optimal steady state is the asymptotically stable solution of the closed-loop system under terminal constraint MPC. Its region of attraction is the feasible set. (Rawlings, Bonn´e, Jørgensen, Venkat, and Jørgensen, 2008)
Rawlings/Angeli/Bates
Economic MPC
25 / 94
Example 1. Single input–single output system
G (s) =
−0.2623 + 59.2s + 1
60s 2
Sample time T = 10 sec Input constraint, −1 ≤ u ≤ 1 Setpoint ysp = 0.25
Qy = 1, R = 0, S = 10−3 Horizon length N = 80 Periodic disturbance d = 2 with Gd = G and exact measurement
Rawlings/Angeli/Bates
Economic MPC
26 / 94
Disturbance estimation As the estimated disturbance changes with time, the setpoint changes between reachable and unreachable. xs
ds ≥ G
0 ≤ ds ≤ G
xsp
0
1 us 0 ≤ us ≤ 1 ds ≤ 0
Rawlings/Angeli/Bates
Economic MPC
27 / 94
Disturbance estimation As the estimated disturbance changes with time, the setpoint changes between reachable and unreachable. xs
xsp
ds ≥ G
0 ≤ ds ≤ G
xsp
0
xs (k)
1 us 0 ≤ us ≤ 1
0 k
ds ≤ 0
Rawlings/Angeli/Bates
ˆ d(k)
Economic MPC
27 / 94
0.3 0.2 0.1 y
0 -0.1 -0.2 -0.3 0
50
100 150 200 250 300 350 400 Time (sec)
setpoint target (ys )
y (sp-MPC) y (targ-MPC)
1 0.5 u
0 -0.5 -1 0
50
100 150 200 250 300 350 400 Time (sec)
target (us ) u(sp-MPC) Rawlings/Angeli/Bates
u(targ-MPC) Economic MPC
28 / 94
Summary of Example 1
Performance Measure
targ-MPC (×10−3 )
sp-MPC (×10−3 )
∆(index)%
Vu Vy V
1.7 × 10−2 6.98 7.00
2.2 × 10−6 3.27 3.27
99.99 53 53
Vu =
Vy =
T −1 1 X |u(k) − usp |2R + |u(k) − u(k − 1)|2S T
1 T
0 T −1 X
|y (k) − ysp |Q 2 y
0
V = Vu + Vy Rawlings/Angeli/Bates
Economic MPC
29 / 94
Example 2. Two input–two output system with noise
" G (s) =
1.5 (s+2)(s+1) 0.5 (s+0.5)(s+1)
0.75 (s+5)(s+2) 2 (s+2)(s+3)
#
Sample time T = 0.25 sec Input constraints −0.5 ≤ u1 , u2 ≤ 0.5 Setpoint ysp = [0.337 0.34]0 Qy = 5I , R = I , S = I Horizon length N = 80 Periodic disturbance d = ±[0.03 − 0.03]0 with Gd = G and measurement and state noise
Rawlings/Angeli/Bates
Economic MPC
30 / 94
0.338 0.337 0.336 0.335 y1 0.334 0.333 0.332
setpoint y1 (sp-MPC) y1 (targ-MPC)
0.331 0.33 0
5
10 15 Time (sec)
20
25
0.342 0.341 0.34 0.339 y2 0.338 0.337 setpoint y2 (sp-MPC) y2 (targ-MPC)
0.336 0.335 0 Rawlings/Angeli/Bates
5
10 15 Time (sec)
20
Economic MPC
25 31 / 94
0.5
u1 (targ-MPC) u1 (sp-MPC)
0.45
u1
0.5
u1 0.45 0
5
10 15 Time (sec)
20
25 -0.45 -0.465
u2
-0.48 u2 (targ-MPC) u2 (sp-MPC) -0.45 u2
-0.465 -0.48 0
Rawlings/Angeli/Bates
5
10 15 Time (sec)
20
Economic MPC
25 32 / 94
y1s 0.3 setpoint target (y1s ) db1
0.2 0.1 0
db1
-0.1 0
5
10 15 Time (sec)
20
25
y2s 0.3 setpoint target (y2s ) db2
0.2
0.1
db2
0 0
Rawlings/Angeli/Bates
5
10 15 Time (sec)
20
Economic MPC
25 33 / 94
Summary of Example 2
Performance Measure
targ-MPC (×10−4 )
sp-MPC (×10−4 )
∆(index)%
Vu Vy V
3.32 1.63 4.95
2.10 0.04 2.14
37 98 57
Rawlings/Angeli/Bates
Economic MPC
34 / 94
Economic MPC: motivating the idea
Profit
-4
Rawlings/Angeli/Bates
-2 0 Input (u)
2
4
Economic MPC
-4
-2
0
4 2 State (x)
35 / 94
Economic MPC: motivating the idea
Profit
-4
Rawlings/Angeli/Bates
-2 0 Input (u)
2
4
Economic MPC
-4
-2
0
4 2 State (x)
35 / 94
Economic MPC definition (with terminal constraint)
Economic stage cost: `(x, u)
Rawlings/Angeli/Bates
Economic MPC
36 / 94
Economic MPC definition (with terminal constraint)
Economic stage cost: `(x, u) Optimization: min VN,e (x, u) = u
N−1 X
`(x(k), u(k))
k=0
+ x = f (x, u) x(0) = x (x(k), u(k)) ∈ Z k ∈ [0 : N − 1] subject to x(N) = xs Control law: u = κN,e (x) Admissible set: XN,e
Rawlings/Angeli/Bates
Economic MPC
36 / 94
(1)
Example
x + = Ax + Bu 0.857 0.884 8.565 A= B= −0.0147 −0.0151 0.88418 Input constraint: −1 ≤ u ≤ 1 `(x, u) = α0 x + β 0 u 0 α = −3 −2 β = −2
Rawlings/Angeli/Bates
`t (x, u) = |x − xs |2Q + |u − us |2R Q = 2I2 R=2 0 xs = 60 0 us = 1
Economic MPC
37 / 94
10 8 6 x2
4 2 0 -2 60
Rawlings/Angeli/Bates
65 70 targ-MPC x1
75
Economic MPC
80
85
38 / 94
10 8 6 x2
4 2 0 -2 60
Rawlings/Angeli/Bates
65 70 targ-MPC x1
75
Economic MPC
80 85 eco-MPC
38 / 94
80
targ-MPC
State 1
75 70 65 60
State 2
55 10 8 6 4 2 0 -2
0
2
4
6
8
10
12
14
targ-MPC
0
2
4
6
8
10
12
14
1 Input
0.5 0 -0.5 -1 0
2
4
6
8
10
targ-MPC 12
14
Time
Rawlings/Angeli/Bates
Economic MPC
39 / 94
State 1
90 85 80 75 70 65 60 55
State 2
10 8 6 4 2 0 -2
targ-MPC eco-MPC
0
2
4
6
8
10
12
14
targ-MPC eco-MPC
0
2
4
6
8
0
2
4
6
8
10
12
14
1 Input
0.5 0 -0.5 -1 10
targ-MPC eco-MPC 12
14
Time
Rawlings/Angeli/Bates
Economic MPC
39 / 94
Closed-loop performance measures Profitability: I
Average asymptotic cost relative to steady state T 1 X `(x(k), u(k)) − `(xs , us ) T →∞ T
lim
k=0
I
Net cost relative to steady state ∞ X k=0
`(x(k), u(k)) − `(xs , us )
Stability: I
Asymptotic convergence to optimal steady state lim (x(k), u(k)) = (xs , us )
k→∞
Rawlings/Angeli/Bates
Economic MPC
40 / 94
How can profitability and stability be opposing goals? We consider a nonlinear constant volume isothermal CSTR I I
State: CA Input: CAf
The following reactions take place: r = kcA2
A→B Economic stage cost:
`(x, u) = −CB Input constraints over horizon of N: 0 ≤ u(k) ≤ 3
Rawlings/Angeli/Bates
N 1 X u(k) = 1 N k=0
Economic MPC
41 / 94
Optimal control solution
Optimal u and x 0.5
4
0.4 3 0.3
u
2
x 0.2
1 0.1
0
0 0
20
40
t
60
Rawlings/Angeli/Bates
80
100
Economic MPC
42 / 94
Optimal control solution
Production rate, RB = kcA2
Optimal u and x 4
0.5
0.3
0.4
0.25
cAs
3 0.2 0.3
u
2
cA2
x
0.15
0.2 0.1
hcA2 i
1 0.1 0.05 0
0 0
20
40
t
60
Rawlings/Angeli/Bates
80
100
2 cAs
0 0
Economic MPC
20
40
t
60
80
42 / 94
100
Average economic performance of EMPC
When considering economic optimization as the objective of control, average economic performance is the more natural performance measure. In tracking MPC, average closed-loop economic performance is guaranteed indirectly, via stability T 1 X `(x(k), u(k)) = `(xs , us ) lim (x(k), u(k)) = (xs , us ) ⇒ lim k→∞ T →∞ T k=0
Several methods are available for stabilizing tracking MPC including the addition of a terminal equality constraint, x(N) = xs , and a terminal penalty
Rawlings/Angeli/Bates
Economic MPC
43 / 94
What does a terminal constraint accomplish for EMPC?
Theorem: Average economic performance of EMPC The average asymptotic cost of the closed-loop system x + = f (x, κN,e (x)) satisfies
PT lim sup T →+∞
Rawlings/Angeli/Bates
k=0 `(x(k), u(k))
T +1
≤ `(xs , us )
Economic MPC
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What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state
Rawlings/Angeli/Bates
Economic MPC
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What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I
EMPC is stable under these conditions
Rawlings/Angeli/Bates
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What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
Rawlings/Angeli/Bates
Economic MPC
45 / 94
What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state? ∞ X k=0
Rawlings/Angeli/Bates
`(x(k), u(k)) − `(xs , us )
Economic MPC
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What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state? ∞ X k=0 I
`(x(k), u(k)) − `(xs , us )
Bounded above because VN0 (x) is bounded on XN .
Rawlings/Angeli/Bates
Economic MPC
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What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state? ∞ X k=0 I I
`(x(k), u(k)) − `(xs , us )
Bounded above because VN0 (x) is bounded on XN . Not bounded below because the controller can outperform the best steady state on average
Rawlings/Angeli/Bates
Economic MPC
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What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state? ∞ X k=0 I I
I
`(x(k), u(k)) − `(xs , us )
Bounded above because VN0 (x) is bounded on XN . Not bounded below because the controller can outperform the best steady state on average No proven relationship between closed-loop cost for tracking MPC vs. EMPC from a given initial state
Rawlings/Angeli/Bates
Economic MPC
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Industrial simulation example F200
F4 , T3
Separator L2
Cooling water T200
T201 Condenser
Condensate F5
P100 LT Steam F100
T100
Evaporator
Condensate
LC
F3
F2
Feed F1 , X1 , T1
Product X2 , T2
Evaporator system (Newell and Lee, 1989, Ch. 2) Rawlings/Angeli/Bates
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Evaporator system Measurements: product composition X2 and operating pressure P2 Inputs: steam pressure P100 , cooling water flow rate F200 The economic stage cost is the operating cost for electricity, steam and cooling water (Wang and Cameron, 1994; Govatsmark and Skogestad, 2001). J = 1.009(F2 + F3 ) + 600F100 + 0.6F200
Rawlings/Angeli/Bates
Economic MPC
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Evaporator system Measurements: product composition X2 and operating pressure P2 Inputs: steam pressure P100 , cooling water flow rate F200 The economic stage cost is the operating cost for electricity, steam and cooling water (Wang and Cameron, 1994; Govatsmark and Skogestad, 2001). J = 1.009(F2 + F3 ) + 600F100 + 0.6F200
We consider the process subject to disturbances in feed flow rate F1 , Feed composition C1 , Circulating flow rate F3 , feed temperature T1 and cooling water inlet temperature T200
Rawlings/Angeli/Bates
Economic MPC
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Evaporator system Measurements: product composition X2 and operating pressure P2 Inputs: steam pressure P100 , cooling water flow rate F200 The economic stage cost is the operating cost for electricity, steam and cooling water (Wang and Cameron, 1994; Govatsmark and Skogestad, 2001). J = 1.009(F2 + F3 ) + 600F100 + 0.6F200
We consider the process subject to disturbances in feed flow rate F1 , Feed composition C1 , Circulating flow rate F3 , feed temperature T1 and cooling water inlet temperature T200 We consider both tracking MPC and EMPC with a terminal state constraint Rawlings/Angeli/Bates
Economic MPC
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14000 12000 Cost ($/h)
Disturbance P2 (kPa)
0 -1 -2 -3 -4 -5 -6 -7 -8 -9
0
20
40 60 time (min)
80
0
20
40 60 time (min)
80
100
0
20
40 60 time (min)
80
100
80
100
220 Input F200 (kg/min)
Input P100 (kPa)
6000 2000
100
350 300 250 200 150 100
0
20
40 60 time (min)
80
200 180 160 140 120 100
100
70
50
60
Output P2 (kPa)
Output X2(%)
8000 4000
400
50 40 30 20
10000
0
20
Rawlings/Angeli/Bates
40 60 time (min)
80
100
eco-MPC track-MPC
45 40 35 30
0
20
Economic MPC
40 60 time (min)
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12000
F1 T1
Cost ($/h)
Disturbance
50 45 40 35 30 25 20 15 10
T100 0 10 20 30 40 50 60 70 80 90 time (min)
6000 0 10 20 30 40 50 60 70 80 90 time (min)
400 Input F200 (kg/min)
Input P100 (kPa)
8000
4000
400 350 300 250 200 150 100
10000
360 320 280 240 200
0 10 20 30 40 50 60 70 80 90 time (min)
0 10 20 30 40 50 60 70 80 90 time (min)
36
Output P2 (kPa)
Output X2(%)
40
32 28 24 20
0 10 20 30 40 50 60 70 80 90 time (min)
Rawlings/Angeli/Bates
56
eco-MPC track-MPC
54 52 50 0 10 20 30 40 50 60 70 80 90 time (min)
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Evaporator system closed-loop economics
Performance comparison under economic and tracking MPC
Disturbance Measured Unmeasured
Rawlings/Angeli/Bates
Avg. operating cost ×10−6 ($/ hr) eco-MPC track-MPC ∆ (%) 5.89 5.80
5.97 6.15
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2.2 6.2
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Asymptotic stability and EMPC
Steady operation often desired by practitioners I I
Equipment not designed for strongly unsteady operation Operator acceptance issue for unsteady operation
Rawlings/Angeli/Bates
Economic MPC
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Asymptotic stability and EMPC
Steady operation often desired by practitioners I I
Equipment not designed for strongly unsteady operation Operator acceptance issue for unsteady operation
Stability analysis I
I
Check that stability is consistent with the process model and control objectives Or modify the control objectives (stage cost) given the process model
Rawlings/Angeli/Bates
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Stabilizing assumption for EMPC System: x + = f (x, u) Supply rate: s(x, u)
Storage: λ(x)
Dissipation
Assumption: Dissipativity The system x + = f (x, u) is dissipative with respect to the supply rate s : Z → R if there exists a function λ : X → R such that: λ(f (x, u)) − λ(x) ≤ s(x, u) for all (x, u) ∈ Z. If ρ : X → R≥0 positive definite exists such that: λ(f (x, u)) − λ(x) ≤ −ρ(x) + s(x, u) then the system is said to be strictly dissipative. Rawlings/Angeli/Bates
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Optimality of steady-state operation Optimal operation at steady-state The system x + = f (x, u) is optimally operated at steady-state if for all trajectories (x, u), such that (x(k), u(k)) ∈ Z for all k ∈ N it holds that PT −1 lim inf
T →+∞
Rawlings/Angeli/Bates
k=0
`(x(k), u(k)) ≥ `(xs , us ) T
Economic MPC
53 / 94
Optimality of steady-state operation Optimal operation at steady-state The system x + = f (x, u) is optimally operated at steady-state if for all trajectories (x, u), such that (x(k), u(k)) ∈ Z for all k ∈ N it holds that PT −1 lim inf
T →+∞
k=0
`(x(k), u(k)) ≥ `(xs , us ) T
Suboptimal operation off steady-state A system optimally operated at steady-state is suboptimally operated off steady-state if for all trajectories (x, u), such that (x(k), u(k)) ∈ Z it holds that either PT −1 k=0 `(x(k), u(k)) lim inf > `(xs , us ) T →+∞ T or lim inf |x(k) − xs | = 0. k→+∞
Rawlings/Angeli/Bates
Economic MPC
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Dissipativity and Optimality
Dissipativity is closely related to optimal operation at steady-state 1 2
Dissipativity ⇒ Optimal operation at steady-state
Strict Dissipativity ⇒ Sub-optimal operation off steady-state
Rawlings/Angeli/Bates
Economic MPC
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Dissipativity and Optimality
Dissipativity is closely related to optimal operation at steady-state 1 2 3
Dissipativity ⇒ Optimal operation at steady-state
Strict Dissipativity ⇒ Sub-optimal operation off steady-state
Gap between Dissipativity and Optimal steady state operation: ∀ x, all trajectories (x, u) such that (x(k), u(k)) ∈ Z and x(0) = x satisfy inf
T ≥1
T −1 X k=0
`(x(k), u(k)) − `(xs , us ) > −∞
investigated by Mueller et al. (2013)
Rawlings/Angeli/Bates
Economic MPC
54 / 94
Dissipativity: an example
Consider the following system x + = αx + (1 − α)u,
α ∈ [0, 1)
With the following nonconvex stage cost `(x, u) = (x + u/3)(2u − x) + (x − u)4
1
Strong duality requires constant λ such that λ0 f (x, u) − λ0 x ≤ `(x, u). Rawlings/Angeli/Bates
Economic MPC
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Dissipativity: an example
Consider the following system x + = αx + (1 − α)u,
α ∈ [0, 1)
With the following nonconvex stage cost `(x, u) = (x + u/3)(2u − x) + (x − u)4 This system and stage cost are dissipative for α ∈ [ 12 , 1], but are not strongly dual1 for any α. What does this tell us about the behavior of EMPC?
1
Strong duality requires constant λ such that λ0 f (x, u) − λ0 x ≤ `(x, u). Rawlings/Angeli/Bates
Economic MPC
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0.2 0.15 0.1 State
0.05 0 -0.05 -0.1 -0.15 -0.2 0
2
4
6
α = 0.2
8
10 Time
12
α = 0.4
14
16
18
α = 0.6
0.3 0.2
Input
0.1 0 -0.1 -0.2 -0.3 0
2
α = 0.2 Rawlings/Angeli/Bates
4
6
8
10 Time
α = 0.4
12
14
16
18
α = 0.6 Economic MPC
56 / 94
0.05
α = .2 α = .4 α = .6
0.04
Stage Cost
0.03 0.02 0.01 0 -0.01 -0.02 -0.03 0
2
4
6
8
10 Time
12
14
16
18
EMPC controller with N = 10, terminal equality constraint Closed-loop EMPC is stable for α ≥ 1/2
Closed-loop EMPC is unstable for α < 1/2 and outperforms the best steady state on average Dissipativity is a tighter condition for EMPC stability than strong duality Rawlings/Angeli/Bates
Economic MPC
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u
α u = x/2
(b) x
(a)
u = −3x
Qualitative picture of L0 : global minima (a),(b); α-dependent period-2 solution (black dots). Adapted from Angeli et al. (2012).
Rawlings/Angeli/Bates
Economic MPC
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Stability theorem for EMPC
Theorem: Stability of EMPC If the system x + = f (x, κN,e (x)) is strictly dissipative with respect to the supply rate s(x, u) = `(x, u) − `(xs , us ) then xs is an asymptotically stable equilibrium point of the closed-loop system with region of attraction XN,e . Nominal average asymptotic performance not worse than steady operation is always implied by stability
Rawlings/Angeli/Bates
Economic MPC
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Sketch of proof Lyapunov-based proof, with rotated stage cost: L(x, u) := `(x, u) − `(xs , us ) + λ(x) − λ(f (x, u))
Rawlings/Angeli/Bates
Economic MPC
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Sketch of proof Lyapunov-based proof, with rotated stage cost: L(x, u) := `(x, u) − `(xs , us ) + λ(x) − λ(f (x, u)) Notice that N−1 X
L(x(k), u(k))
k=0
=
N−1 X k=0
`(x(k), u(k)) + λ(x(k)) − λ(x(k + 1)) − `(xs , us )
= λ(x(0)) − λ(x(N)) − N`(xs , us ) +
N−1 X
`(x(k), u(k))
k=0
Optimal control is unaffected by cost rotation. Rawlings/Angeli/Bates
Economic MPC
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Sketch of proof (continued) Under dissipativity assumption rotated cost fulfills standard MPC conditions (positive semi-definite): L(x, u) ≥ 0 In addition, under strict dissipativity, the rotated cost is positive definite: L(x, u) ≥ ρ(x)
Rawlings/Angeli/Bates
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Sketch of proof (continued) Under dissipativity assumption rotated cost fulfills standard MPC conditions (positive semi-definite): L(x, u) ≥ 0 In addition, under strict dissipativity, the rotated cost is positive definite: L(x, u) ≥ ρ(x) Cost-to-go can be used as a candidate Lyapunov function V (x) := min u
N−1 X
L(x(k), u(k))
k=0
subject to initial, terminal and dynamic constraints. Rawlings/Angeli/Bates
Economic MPC
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Sketch of proof (continued) Recursive feasibility: Feasibility of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: x = (x(1), x(2), . . . , x(N − 1), xs ) u = (u(1), . . . , u(N − 1), us ).
Rawlings/Angeli/Bates
Economic MPC
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Sketch of proof (continued) Recursive feasibility: Feasibility of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: x = (x(1), x(2), . . . , x(N − 1), xs ) u = (u(1), . . . , u(N − 1), us ). Along closed-loop solutions: V (x + ) ≤ V (x) − ρ(x). Hence, ρ(x(k)) → 0 as k → +∞ and convergence to equilibrium follows by compactness of Z. Rawlings/Angeli/Bates
Economic MPC
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Asymptotic averages As convergence to equilibrium is not always enforced, asymptotic averages of output variables need not equal their limit (which in fact may fail to exist) Possibility of asymptotic average constraints different from pointwise in time constraints (pointwise in time constraints are typically more stringent)
Rawlings/Angeli/Bates
Economic MPC
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Asymptotic averages As convergence to equilibrium is not always enforced, asymptotic averages of output variables need not equal their limit (which in fact may fail to exist) Possibility of asymptotic average constraints different from pointwise in time constraints (pointwise in time constraints are typically more stringent) Definition of asymptotic average: Av [v ] = {w : ∃ {Tn }+∞ n=1 : Tn → +∞ as n → +∞ PTn −1 k=0 v (k) and lim =w } n→+∞ Tn Asymptotic average need not be a singleton: 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, . . . Notice that {1/3, 1/2} ⊂ Av[v ] ⊂ [1/3, 1/2]. Rawlings/Angeli/Bates
Economic MPC
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Average constraints Goals: Modify economic MPC algorithm to guarantee: (x(k), u(k)) ∈ Z for all k ∈ N; Av[h(x, u)] ⊂ Y 3 h(xs , us ); Recursive feasibility
Rawlings/Angeli/Bates
Economic MPC
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Average constraints Goals: Modify economic MPC algorithm to guarantee: (x(k), u(k)) ∈ Z for all k ∈ N; Av[h(x, u)] ⊂ Y 3 h(xs , us ); Recursive feasibility Remarks: 1
For technical reasons Y is assumed to be convex.
2
Average constraint does not imply limits on averages computed on finite time windows.
For the following: At each time t let variables z(k) and v (k) denote virtual (predicted) variables, and x(k), u(k) (k ≤ t) actual variables Rawlings/Angeli/Bates
Economic MPC
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Economic MPC with average constraints A modified receding-horizon strategy: min v,z
N−1 X
`(z(k), v (k))
k=0
subject to: (z(k), v (k)) ∈ Z ∀ k ∈ {0, 1, . . . , N − 1}
z(k + 1) = f (z(k), v (k)) ∀ k ∈ {0, 1, . . . , N − 1} z(0) = x(t) N−1 X k=0
z(N) = xs
h(z(k), v (k)) ∈ Yt
where: Yt+1 = Yt ⊕ Y {h(x(t), u(t))}
Y0 = NY ⊕ Y00
with ⊕, denoting set addition and subtraction, and Y00 is an arbitrary convex compact set. Rawlings/Angeli/Bates
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Properties of the algorithm Time-varying state-feedback Recursive feasibility follows by standard argument: Feasibility of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: x = (x(1), x(2), . . . , x(N − 1), xs ) u = (u(1), . . . , u(N − 1), us ).
Rawlings/Angeli/Bates
Economic MPC
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Properties of the algorithm Time-varying state-feedback Recursive feasibility follows by standard argument: Feasibility of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: x = (x(1), x(2), . . . , x(N − 1), xs ) u = (u(1), . . . , u(N − 1), us ). Additional constraint does not limit feasibility region (Y0 is arbitrarily large) Asymptotic average constraints are guaranteed Average performance not worse than best steady-state Possibility of replacing terminal constraint with terminal penalty function Rawlings/Angeli/Bates
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Working principle Show by induction: Yt = tY + Y0 {
t−1 X
h(x(k), u(k))}
k=0
Hence: N−1 X
h(z(k), v (k)) +
k=0
Rawlings/Angeli/Bates
t−1 X k=0
h(x(k), u(k)) ∈ tY ⊕ Y0
Economic MPC
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Working principle Show by induction: Yt = tY + Y0 {
t−1 X
h(x(k), u(k))}
k=0
Hence: N−1 X
h(z(k), v (k)) +
k=0
t−1 X k=0
h(x(k), u(k)) ∈ tY ⊕ Y0
Taking tn → +∞ so that limit exists: Ptn −1 k=0 h(x(k), u(k)) = lim n→+∞ tn PN−1 Ptn −1 k=0 h(z(k), v (k)) + k=0 h(x(k), u(k)) = lim ∈Y n→+∞ tn Rawlings/Angeli/Bates
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Dissipativity for averagely constrained systems
Problem: Asymptotic convergence of averagely constrained MPC Sufficient conditions, possibility of relaxing dissipativity?
Dissipativity taking into account average constraints Consider the modified supply function: ¯ T [Ah(x, u) − b] sa (x, u) = `(x, u) − `(xs , us ) + λ provided: Y = {y : Ay ≤ b}
Rawlings/Angeli/Bates
Economic MPC
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Dissipativity and Convergence for Averagely Constrained EMPC Optimal steady-state operation If a system is dissipative with respect to the supply rate sa (x, u) for some ¯ then every feasible solution fulfills: non-negative λ lim inf
T →+∞
Rawlings/Angeli/Bates
T −1 X k=0
`(x(k), u(k)) ≥ `(xs , us ) T
Economic MPC
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Dissipativity and Convergence for Averagely Constrained EMPC Optimal steady-state operation If a system is dissipative with respect to the supply rate sa (x, u) for some ¯ then every feasible solution fulfills: non-negative λ lim inf
T →+∞
T −1 X k=0
`(x(k), u(k)) ≥ `(xs , us ) T
Convergence of Averagely Constrained EMPC If a system is strictly dissipative with respect to the supply rate sa (x, u) for ¯ then closed-loop solutions of Averagely Constrained some non-negative λ EMPC asymptotically approach the best steady-state.
Rawlings/Angeli/Bates
Economic MPC
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Enforcing convergence in Economic MPC
By modifying the economic stage cost `(x, u) as: ˜ u) = `(x, u) + γ(|x − xs | + |u − us |) `(x, in order to recover dissipativity or strong duality
Rawlings/Angeli/Bates
Economic MPC
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Enforcing convergence in Economic MPC
By modifying the economic stage cost `(x, u) as: ˜ u) = `(x, u) + γ(|x − xs | + |u − us |) `(x, in order to recover dissipativity or strong duality By adding an auxiliary average constraint, such as: Av[|x − xs |2 ] ⊂ {0} or: Av[|xi − xsi |2 ] ⊂ {0}
Rawlings/Angeli/Bates
i = 1 . . . n;
Economic MPC
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Case study: a reactor with parallel reactions The reactions: R → P1
R → P2
The state-space model: x˙ 1 = 1 − 104 x12 e −1/x3 − 400x1 e −0.55/x3 − x1 x˙ 2 = 104 x12 e −1/x3 − x2 x˙ 3 = u − x3
x1 is concentration of R, x2 is concentration of P1 (the desired product), x3 is the temperature and u is the heat flux. P2 is the waste product (not modeled).
Rawlings/Angeli/Bates
Economic MPC
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Case study: a reactor with parallel reactions The reactions: R → P1
R → P2
The state-space model: x˙ 1 = 1 − 104 x12 e −1/x3 − 400x1 e −0.55/x3 − x1 x˙ 2 = 104 x12 e −1/x3 − x2 x˙ 3 = u − x3
x1 is concentration of R, x2 is concentration of P1 (the desired product), x3 is the temperature and u is the heat flux. P2 is the waste product (not modeled). Goal: maximize x2 , viz.: `(x, u) = −x2 ; Known that best performance is not at steady state Rawlings/Angeli/Bates
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Closed-loop simulations
0.4 0.3 u 0.2 0.1 0
2
4
6
8
10
0.7 0.6 0.5 0.4 x1 0.3 0.2 0.1 00
0.2
IC 1 IC 2 IC 3
2
4
6
8
10
2
4
6
8
10
0.2 0.15
x2
x3 0.1 00
2
4
6
8
10
0.05 0
Closed-loop input and state profiles for economic MPC with a convex term and different initial states
Rawlings/Angeli/Bates
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Closed-loop simulations (2)
0.4 0.3 u 0.2 0.1 0
2
4
6
8
10
0.7 0.6 0.5 0.4 x1 0.3 0.2 0.1 00
0.2
IC 1 IC 2 IC 3
2
4
6
8
10
2
4
6
8
10
0.2 0.15
x2
x3 0.1 00
2
4
6
8
10
0.05 0
Closed-loop input and state profiles for economic MPC with a convergence constraint and different initial states
Rawlings/Angeli/Bates
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Outperforming best steady-state
Terminal constraint instrumental in: 1 2
guaranteeing recursive feasibility providing bound to asymptotic performance
Any feasible trajectory may be used as a terminal constraint
Idea: Replace best equilibrium by best feasible periodic solution of given period
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Best feasible periodic solution Let xs , us be: xs = [xs (0), xs (1), . . . , xs (Q − 1)] us = [us (0), us (1), . . . , us (Q − 1)], and assume that they belong to: arg min x,u
Q−1 X
`(x(k), u(k))
k=0
subject to: x(k + 1) = f (x(k), u(k))
k ∈ {0, 1, . . . , Q − 2}
x(0) = f (x(Q − 1), u(Q − 1)) (x(k), u(k)) ∈ Z
k ∈ {0, 1, . . . , Q − 1}
We call xs , us an optimal Q-periodic solution. Rawlings/Angeli/Bates
Economic MPC
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EMPC with periodic terminal constraint
Solve at each time t the following optimization problem: min v,z
N−1 X
`(z(k), v (k))
k=0
subject to: (z(k), v (k)) ∈ Z ∀ k ∈ {0, 1, . . . , N − 1} z(k + 1) = f (z(k), v (k)) ∀ k ∈ {0, 1, . . . , N − 1} z(0) = x(t)
Rawlings/Angeli/Bates
z(N) = xs (t mod Q)
Economic MPC
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Features of the algorithm
Q periodic state feedback Recursive feasibility: feasibility at time t of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: (x(1), . . . , x(N−1), xs (t mod Q))
(u(1), . . . , u(N−1), us (t mod Q))
at time t + 1. Possibility of incorporating average constraints
Rawlings/Angeli/Bates
Economic MPC
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Average performance with periodic end constraint Let V be the cost to go: V (t, x) = min v,z
N−1 X
`(z(k), v (k))
k=0
subject to: (z(k), v (k)) ∈ Z ∀ k ∈ {0, 1, . . . , N − 1} z(k + 1) = f (z(k), v (k)) ∀ k ∈ {0, 1, . . . , N − 1} z(0) = x
Rawlings/Angeli/Bates
z(N) = xs (t mod Q)
Economic MPC
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Average performance with periodic end constraint Let V be the cost to go: V (t, x) = min v,z
N−1 X
`(z(k), v (k))
k=0
subject to: (z(k), v (k)) ∈ Z ∀ k ∈ {0, 1, . . . , N − 1} z(k + 1) = f (z(k), v (k)) ∀ k ∈ {0, 1, . . . , N − 1} z(0) = x
z(N) = xs (t mod Q)
Along closed-loop solution: V (t + 1, x(t + 1)) ≤ V (t, x(t))
− `(x(t), u(t)) + `(xs (t mod Q), us (t mod Q))
Rawlings/Angeli/Bates
Economic MPC
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Average performance with periodic end constraint
Taking sums between 0 and T − 1 and dividing by T yields: PT −1 lim sup T →+∞
k=0
`(x(k), u(k)) ≤ T
PQ−1 k=0
`(xs (k), us (k)) Q
Average performance at least as good as optimal Q-periodic solution
Rawlings/Angeli/Bates
Economic MPC
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Average performance with periodic end constraint
Taking sums between 0 and T − 1 and dividing by T yields: PT −1 lim sup T →+∞
k=0
`(x(k), u(k)) ≤ T
PQ−1 k=0
`(xs (k), us (k)) Q
Average performance at least as good as optimal Q-periodic solution Q and N may be different from each other and unrelated The closed-loop system need not be asymptotically stable to the optimal Q periodic solution The optimal Q periodic solution need not be an equilibrium of the closed-loop system
Rawlings/Angeli/Bates
Economic MPC
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Ammonia synthesis (Jain et al., 1985)
Open loop stable equilibria Open-loop periodic forcing improves average production rate 1 2
Improvement of 24% by using sinusoidal inputs u(t) = c + A sin(ωt) Improvement of 25% by using square-waves of different amplitudes, average and duty cycle
Rawlings/Angeli/Bates
Economic MPC
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Simulation of EMPC with terminal steady-state constraint
Improvement of 30 % in Ammonia production rate
Rawlings/Angeli/Bates
Economic MPC
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Best Q-periodic solutions
Discretization Ts = 0.1 Best Q-periodic solution for Q = 16 Average production rate: 50 % better than best square wave Rawlings/Angeli/Bates
Economic MPC
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Closed-loop with Q periodic terminal constraint
Chaotic regime: 25 % better than periodic solution Overall ≈ 140 % improvement over best steady state Rawlings/Angeli/Bates
Economic MPC
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Terminal penalty formulations Just as with tracking MPC, we can expand the feasible set XN by replacing the terminal equality constraint with a terminal set constraint and a terminal penalty Admissible set: XN,p ={x ∈ X | ∃ u such that the trajectory (x(k), u(k)) satisfies (x(k), u(k)) ∈ Z k ∈ I0:N−1 ,
x(N) ∈ Xf }
Objective function: VN,p (x, u) =
N−1 X
`(x(k), u(k)) + Vf (x(N))
k=0
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Terminal penalty in EMPC
Assumption: Terminal penalty There exists a terminal region control law κf : Xf → U such that Vf (f (x, κf (x))) ≤ Vf (x) − `(x, κf (x)) + `(xs , us ) (x, κf (x)) ∈ ZN,p
∀x ∈ Xf
This assumption is identical in form to the tracking case, but `(x, u) has changed Vf (x) need not be positive definite with respect to xs
Rawlings/Angeli/Bates
Economic MPC
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Terminal penalty in EMPC
Theorem: EMPC stability with terminal penalty If the system x + = f (x, κN,p (x)) is strictly dissipative with respect to the supply rate: s(x, u) = `(x, u) − `(xs , us ) then xs is an asymptotically stable equilibrium point of the closed-loop system with region of attraction XN .
Rawlings/Angeli/Bates
Economic MPC
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Case study: nonlinear CSTR We consider a nonlinear constant volume isothermal CSTR I I
States: P0 , B, P1 , P2 Inputs: inflow concentrations of P0 , B
The following reactions take place: P0 + B −→ P1 P1 + B −→ P2
Rawlings/Angeli/Bates
Economic MPC
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Case study: nonlinear CSTR We consider a nonlinear constant volume isothermal CSTR I I
States: P0 , B, P1 , P2 Inputs: inflow concentrations of P0 , B
The following reactions take place: P0 + B −→ P1 P1 + B −→ P2 Economic stage cost: `(x, u) = −CP1 The controller stage cost is modified according to: ˜ u) = `(x, u) + |x − xs |2 + |u − us |2 `(x, Q R Rawlings/Angeli/Bates
Economic MPC
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Representative open-loop trajectories 2
6
1.5
4
x1 1
x2 2
0.5 00
5
10
15
1.5
00 0.8
1 x4 0.4
x3
5
10
15
5 10 Time (t)
15
Eco R =0 Str. dual track
0.5 00
5 10 Time (t)
15
00
Open-loop state profiles with different cost functions and arbitrary initial state.
Rawlings/Angeli/Bates
Economic MPC
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Average and net profits
Case Unstable Stable
Economic Dissipative Strongly dual Tracking
avg profit
net profit
0.46 0.38 0.38 0.38
∞ 2.6 1.6 1.0
Average profit and net profit for open-loop system with different stage costs.
All stable schemes have about the same average profit (large N) Net profit decreases with increase in tracking term weight
Rawlings/Angeli/Bates
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Conclusions
The economic objective function of EMPC causes novel behavior I
EMPC may be unstable where MPC is stable
Rawlings/Angeli/Bates
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Conclusions
The economic objective function of EMPC causes novel behavior I I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state
Rawlings/Angeli/Bates
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Rawlings/Angeli/Bates
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC I
Terminal penalty formulation
Rawlings/Angeli/Bates
Economic MPC
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC I I
Terminal penalty formulation Average constraints
Rawlings/Angeli/Bates
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC I I I
Terminal penalty formulation Average constraints Periodic terminal constraints
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction.
Rawlings/Angeli/Bates
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009)
Rawlings/Angeli/Bates
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009) Analyzing closed-loop performance
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009) Analyzing closed-loop performance I
What can be proven about net closed-loop performance of tracking MPC relative to EMPC?
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009) Analyzing closed-loop performance I
I
What can be proven about net closed-loop performance of tracking MPC relative to EMPC? What is observed about differences in net closed-loop performance in simulations?
Rawlings/Angeli/Bates
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009) Analyzing closed-loop performance I
I
I
What can be proven about net closed-loop performance of tracking MPC relative to EMPC? What is observed about differences in net closed-loop performance in simulations? What model, stage cost and disturbance characteristics cause large performance differences between MPC and EMPC?
Rawlings/Angeli/Bates
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Open research issues
Tuning EMPC and robustness I
For nondissipative process/stage costs, how should the stage cost be modified?
Rawlings/Angeli/Bates
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Open research issues
Tuning EMPC and robustness I
I
For nondissipative process/stage costs, how should the stage cost be modified? How robust is EMPC to model errors and disturbances?
Rawlings/Angeli/Bates
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Open research issues
Tuning EMPC and robustness I
I I
For nondissipative process/stage costs, how should the stage cost be modified? How robust is EMPC to model errors and disturbances? How can economic risk be incorporated into the controller?
Rawlings/Angeli/Bates
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Open research issues
Tuning EMPC and robustness I
I I
For nondissipative process/stage costs, how should the stage cost be modified? How robust is EMPC to model errors and disturbances? How can economic risk be incorporated into the controller?
Computational methods for implementing EMPC; strategies for adapting existing control hierarchies
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Further reading I D. Angeli, R. Amrit, and J. B. Rawlings. On average performance and stability of economic model predictive control. IEEE Trans. Auto. Cont., 57(7):1615–1626, 2012. R. Dorfman, P. Samuelson, and R. Solow. Linear Programming and Economic Analysis. McGraw-Hill, New York, 1958. L. Fagiano and A. R. Teel. Model predictive control with generalized terminal state constraint. In IFAC Conference on Nonlinear Model Predictive Control 2012, pages 299–304, Noordwijkerhout, the Netherlands, August 2012. A. Ferramosca, D. Limon, I. Alvarado, T. Alamo, and E. Camacho. MPC for tracking of constrained nonlinear systems. In IEEE Conference on Decision and Control (CDC), pages 7978–7983, Shanghai, China, 2009. M. S. Govatsmark and S. Skogestad. Control structure selection for an evaporation process. Comput. Chem. Eng., 9:657–662, 2001. L. Gr¨ une. NMPC without terminal constraints. In IFAC Conference on Nonlinear Model Predictive Control 2012, pages 1–13, Noordwijkerhout, the Netherlands, August 2012a. L. Gr¨ une. Economic receding horizon control without terminal constraints. Automatica, 2012b. Accepted for publication. Rawlings/Angeli/Bates
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Further reading II
A. K. Jain, R. R. Hudgins, and P. L. Silveston. Effectiveness factor under cyclic operation of a reactor. Can. J. Chem., 63:166–169, February 1985. M. Mueller, D. Angeli, and F. Allg¨ ower. On convergence of averagely constrained economic MPC and necessity of dissipativity for optimal steady-state operation. In American Control Conference, 2013. R. B. Newell and P. L. Lee. Applied Process Control – A Case Study. Prentice Hall, Sydney, 1989. J. B. Rawlings, D. Bonn´e, J. B. Jørgensen, A. N. Venkat, and S. B. Jørgensen. Unreachable setpoints in model predictive control. IEEE Trans. Auto. Cont., 53(9): 2209–2215, October 2008. F. Wang and I. Cameron. Control studies on a model evaporation process–constrained state driving with conventional and higher relative degree systems. J. Proc. Cont., 4 (2):59–75, 1994.
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Dept. of Electrical and Electronic Engineering, Imperial College London, UK
CDC Meeting Maui, HI December 10-14, 2012 Rawlings/Angeli/Bates
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Outline 1
Introduction to MPC and economics
2
Stability of standard (tracking) MPC
3
Unreachable setpoints and turnpikes
4
Economic MPC
5
Dissipativity
6
Average constraints
7
Periodic terminal constraint
8
Conclusions and open research issues
Rawlings/Angeli/Bates
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Optimizing economics: current industrial practice
Planning and Scheduling 1
Steady State Optimization
Model Update
Validation
Reconciliation
Two layer structure I
Steady-state layer F
F
RTO optimizes steady state model Optimal setpoints passed to dynamic layer
Controller
Plant
Rawlings/Angeli/Bates
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Optimizing economics: current industrial practice
Planning and Scheduling 1
Steady State Optimization
Model Update
Validation
Reconciliation
Two layer structure I
Steady-state layer F
F
I
Controller
Dynamic layer F
F
Plant
Rawlings/Angeli/Bates
RTO optimizes steady state model Optimal setpoints passed to dynamic layer
Economic MPC
Controller tracks the setpoints Linear MPC
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Optimizing economics: current industrial practice
Planning and Scheduling 1
Steady State Optimization
Model Update
Validation
Reconciliation
2
Two layer structure Drawbacks
Controller
Plant
Rawlings/Angeli/Bates
Economic MPC
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Optimizing economics: current industrial practice
Planning and Scheduling 1
Steady State Optimization
Model Update
Validation
Reconciliation
2
Two layer structure Drawbacks I I
I
Controller I
Inconsistent models Re-identify linear model as setpoint changes Time scale separation may not hold Economics unavailable in dynamic layer
Plant
Rawlings/Angeli/Bates
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Steady-state optimization problem definition
Stage cost: `(x, u) Optimization: (xs , us ) = arg min x,u
`(x, u)
subject to: x = f (x, u),
Rawlings/Angeli/Bates
Economic MPC
(x, u) ∈ Z
5 / 94
Tracking MPC
Setpoint Outputs Inputs
← Past
y u
k=0
Future →
One step of a closed-loop MPC trajectory
Rawlings/Angeli/Bates
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Tracking MPC problem definition Stage cost: `t (x, s) = |x(k) − xs |2Q + |u(k) − us |2R + |u(k) − u(k − 1)|2S
Rawlings/Angeli/Bates
Economic MPC
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Tracking MPC problem definition Stage cost: `t (x, s) = |x(k) − xs |2Q + |u(k) − us |2R + |u(k) − u(k − 1)|2S Optimization: min VN (x, u) = u
N−1 X
`t (x(k), u(k))
k=0
+ x = f (x, u) (x(k), u(k)) ∈ Z k ∈ I0:N−1 subject to x(N) = xs x(0) = x Control law: u = κN (x) Admissible set: XN Rawlings/Angeli/Bates
Economic MPC
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Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller!
Rawlings/Angeli/Bates
Economic MPC
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Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
Rawlings/Angeli/Bates
Economic MPC
8 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
x2
k 0
x1
Rawlings/Angeli/Bates
Economic MPC
8 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
x2
k +1 k 0
x1
Rawlings/Angeli/Bates
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8 / 94
Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
x2
k +2 k +1 k 0
x1
Rawlings/Angeli/Bates
Economic MPC
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Why worry about stability for tracking MPC? Unexpected closed-loop behavior A finite horizon objective function may not give a stable controller! How is this possible?
x2
closed-loop trajectory k +2 k +1 k 0
x1
Rawlings/Angeli/Bates
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Infinite horizon solution The infinite horizon ensures stability
Rawlings/Angeli/Bates
Economic MPC
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Infinite horizon solution The infinite horizon ensures stability Open-loop predictions equal to closed-loop behavior
Rawlings/Angeli/Bates
Economic MPC
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Infinite horizon solution The infinite horizon ensures stability Open-loop predictions equal to closed-loop behavior May be difficult to implement
x2
0 k
Φk x1 Rawlings/Angeli/Bates
Economic MPC
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Infinite horizon solution The infinite horizon ensures stability Open-loop predictions equal to closed-loop behavior May be difficult to implement
x2
0
k +1 k
Vk+1 = Vk − L(xk , uk ) Φk x1 Rawlings/Angeli/Bates
Economic MPC
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Infinite horizon solution The infinite horizon ensures stability Open-loop predictions equal to closed-loop behavior May be difficult to implement
x2 Vk+2 = Vk+1 − L(xk+1, uk+1) k +2 0
k +1 k
Vk+1 = Vk − L(xk , uk ) Φk x1 Rawlings/Angeli/Bates
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Terminal constraint solution Adding a terminal constraint ensures stability
Rawlings/Angeli/Bates
Economic MPC
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Terminal constraint solution Adding a terminal constraint ensures stability May cause infeasibility
Rawlings/Angeli/Bates
Economic MPC
10 / 94
Terminal constraint solution Adding a terminal constraint ensures stability May cause infeasibility Open-loop predictions not equal to closed-loop behavior
x2
k 0
Φk x1 Rawlings/Angeli/Bates
Economic MPC
10 / 94
Terminal constraint solution Adding a terminal constraint ensures stability May cause infeasibility Open-loop predictions not equal to closed-loop behavior
x2
k +1 k 0 Vk+1 ≤ Vk − L(xk , uk ) Φk x1 Rawlings/Angeli/Bates
Economic MPC
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Terminal constraint solution Adding a terminal constraint ensures stability May cause infeasibility Open-loop predictions not equal to closed-loop behavior
x2 Vk+2 ≤ Vk+1 − L(xk+1, uk+1) k +2 k +1 k 0 Vk+1 ≤ Vk − L(xk , uk ) Φk x1 Rawlings/Angeli/Bates
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Closed-loop stability of tracking MPC Assumption: Model, cost and admissible set 1
2
The model f (·) and stage cost `(·) are continuous. The admissible set XN contains xs in its interior. There exists a set Xf containing xs in its interior and K∞ -function γ(·) such that VN0 (x) ≤ γ(|x − xs |) for x ∈ Xf .
Rawlings/Angeli/Bates
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Closed-loop stability of tracking MPC Assumption: Model, cost and admissible set 1
2
The model f (·) and stage cost `(·) are continuous. The admissible set XN contains xs in its interior. There exists a set Xf containing xs in its interior and K∞ -function γ(·) such that VN0 (x) ≤ γ(|x − xs |) for x ∈ Xf .
Theorem: Stability of tracking MPC with terminal constraint The steady-state target (xs , us ) is an asymptotically stable equilibrium point of the closed-loop system x + = f (x, κN (x)) with region of attraction XN . Rawlings/Angeli/Bates
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Setpoints and unreachable setpoints
Rawlings/Angeli/Bates
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Setpoints and unreachable setpoints
Consider the steady state of a linear dynamic model with state x, controlled input u, and disturbance w x(k + 1) = Ax(k) + Bu(k) + Bd w (k)
Rawlings/Angeli/Bates
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Setpoints and unreachable setpoints
Consider the steady state of a linear dynamic model with state x, controlled input u, and disturbance w x(k + 1) = Ax(k) + Bu(k) + Bd w (k) xs = (I − A)−1 B us + (I − A)−1 Bd ws | {z } | {z } G
ds
xs = Gus + ds
Rawlings/Angeli/Bates
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Steady states—unconstrained system xs xs = Gus + ds ds1 = 1
ds2 = 0
ds3 = −1
G xsp
us1
us2
us3 us
For an unconstrained system with G 6= 0, any setpoint xsp with any disturbance ds has a corresponding us . Rawlings/Angeli/Bates
Economic MPC
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Constraints and unreachable setpoints xs xs = Gus + ds ds1 = 1
0 ≤ us ≤ 1
ds2 = 0
ds3 = −1
G xsp
0
us1
1 us2 us3
us
For a constrained system, the setpoint xsp may be unreachable for a given disturbance ds . MPC is method of choice for this situation. Rawlings/Angeli/Bates
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Constraints and unreachable setpoints xs xs = Gus + ds 0 ≤ us ≤ 1 ds ≥ G
0 ≤ ds ≤ G
xsp
0
1 us ds ≤ 0
As the estimated disturbance changes with time, the setpoint may change between reachable and unreachable. Rawlings/Angeli/Bates
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What closed-loop behavior is desirable? Fast tracking
xsp x(0) Q R (fast tracking) x∗
x
x(0)
k
Rawlings/Angeli/Bates
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What closed-loop behavior is desirable? Slow tracking
xsp x(0) Q R (slow tracking) x∗
x
x(0)
k
Rawlings/Angeli/Bates
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What closed-loop behavior is desirable? Asymmetric tracking
xsp x(0) Q R (fast tracking) x∗
x
x(0)
k
Rawlings/Angeli/Bates
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Creating a turnpike example
Standard linear quadratic problem x + = Ax + Bu `(x, u) = |Cx − ysp |2Q + |u − usp |2R
Rawlings/Angeli/Bates
Economic MPC
Q > 0, R > 0
19 / 94
Creating a turnpike example
Standard linear quadratic problem x + = Ax + Bu `(x, u) = |Cx − ysp |2Q + |u − usp |2R
Q > 0, R > 0
Choose an inconsistent setpoint A = 1/2
B = 1/4 ys = Gus usp = 0
Rawlings/Angeli/Bates
C =1
Q=1
R=1
G = 1/2 ysp = 2
Economic MPC
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Inconsistent setpoint and optimal steady state (usp, xsp)
Optimal steady state
x
usp = 0 us = 0.8
xsp = 2 xs = 0.4
(us , xs ) G u
Rawlings/Angeli/Bates
Economic MPC
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Optimal control problem
Cost function and dynamic model VN (x, u) =
N−1 X
`(x(k), u(k))
s.t. x + = Ax + Bu,
x(0) = x
k=0
Rawlings/Angeli/Bates
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Optimal control problem
Cost function and dynamic model VN (x, u) =
N−1 X
`(x(k), u(k))
s.t. x + = Ax + Bu,
x(0) = x
k=0
Optimal state and input trajectories min V (x, u) u
Rawlings/Angeli/Bates
u0 (x),
x0 (x)
Economic MPC
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Optimal trajectory: xsp = 2, usp = 0 1 0.5
x
0
N =5
-0.5 -1 0
u
1
1 0.9 0.8 0.7 0.6 0.5 0.4
2
3
4
2 t
3
4
x0 = −1 x0 = 1
0 Rawlings/Angeli/Bates
1
Economic MPC
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Optimal trajectory: xsp = 2, usp = 0 1 0.5
x
N = 30
0 -0.5 -1 0
u
1 0.9 0.8 0.7 0.6 0.5 0.4
5
10
15
20
25
30
10
15 t
20
25
30
x0 = −1 x0 = 1
0 Rawlings/Angeli/Bates
5
Economic MPC
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Optimal trajectory: xsp = 2, usp = 0 1 0.5
x
N = 100
0 -0.5 -1 0
u
20
30
40
50
60
70
80
90 100
1 0.9 x0 = −1 0.8 0.7 x0 = 1 0.6 0.5 0.4 0 10 20
30
40
50 t
60
70
80
90 100
Rawlings/Angeli/Bates
10
Economic MPC
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Introduction to Turnpike Literature
It is exactly like a turnpike paralleled by a network of minor roads.
Rawlings/Angeli/Bates
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Introduction to Turnpike Literature
It is exactly like a turnpike paralleled by a network of minor roads. There is a fastest route between any two points; and if the origin and destination are close together and far from the turnpike, the best route may not touch the turnpike.
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Introduction to Turnpike Literature
It is exactly like a turnpike paralleled by a network of minor roads. There is a fastest route between any two points; and if the origin and destination are close together and far from the turnpike, the best route may not touch the turnpike. But if the origin and destination are far enough apart, it will always pay to get on the turnpike and cover distance at the best rate of travel, even if this means adding a little mileage at either end.
Rawlings/Angeli/Bates
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Introduction to Turnpike Literature
It is exactly like a turnpike paralleled by a network of minor roads. There is a fastest route between any two points; and if the origin and destination are close together and far from the turnpike, the best route may not touch the turnpike. But if the origin and destination are far enough apart, it will always pay to get on the turnpike and cover distance at the best rate of travel, even if this means adding a little mileage at either end. —Dorfman, Samuelson, and Solow (1958, p.331)
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Economic MPC
23 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply Simulations indicate the closed loop is stable
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—challenges for analyzing closed-loop behavior Consider the MPC controller with the stage cost `(x, s) = |x(k) − xsp |2Q + |u(k) − usp |2R + |u(k) − u(k − 1)|2S Sequence of optimal costs is not monotone decreasing Infinite horizon cost is unbounded for all input sequences Optimal cost is not a Lyapunov function for the closed-loop system Standard nominal MPC stability arguments do not apply Simulations indicate the closed loop is stable How can we be sure?
Rawlings/Angeli/Bates
Economic MPC
24 / 94
Unreachable case—stability result (linear model)
Theorem: Asymptotic Stability of Terminal Constraint MPC The optimal steady state is the asymptotically stable solution of the closed-loop system under terminal constraint MPC. Its region of attraction is the feasible set. (Rawlings, Bonn´e, Jørgensen, Venkat, and Jørgensen, 2008)
Rawlings/Angeli/Bates
Economic MPC
25 / 94
Example 1. Single input–single output system
G (s) =
−0.2623 + 59.2s + 1
60s 2
Sample time T = 10 sec Input constraint, −1 ≤ u ≤ 1 Setpoint ysp = 0.25
Qy = 1, R = 0, S = 10−3 Horizon length N = 80 Periodic disturbance d = 2 with Gd = G and exact measurement
Rawlings/Angeli/Bates
Economic MPC
26 / 94
Disturbance estimation As the estimated disturbance changes with time, the setpoint changes between reachable and unreachable. xs
ds ≥ G
0 ≤ ds ≤ G
xsp
0
1 us 0 ≤ us ≤ 1 ds ≤ 0
Rawlings/Angeli/Bates
Economic MPC
27 / 94
Disturbance estimation As the estimated disturbance changes with time, the setpoint changes between reachable and unreachable. xs
xsp
ds ≥ G
0 ≤ ds ≤ G
xsp
0
xs (k)
1 us 0 ≤ us ≤ 1
0 k
ds ≤ 0
Rawlings/Angeli/Bates
ˆ d(k)
Economic MPC
27 / 94
0.3 0.2 0.1 y
0 -0.1 -0.2 -0.3 0
50
100 150 200 250 300 350 400 Time (sec)
setpoint target (ys )
y (sp-MPC) y (targ-MPC)
1 0.5 u
0 -0.5 -1 0
50
100 150 200 250 300 350 400 Time (sec)
target (us ) u(sp-MPC) Rawlings/Angeli/Bates
u(targ-MPC) Economic MPC
28 / 94
Summary of Example 1
Performance Measure
targ-MPC (×10−3 )
sp-MPC (×10−3 )
∆(index)%
Vu Vy V
1.7 × 10−2 6.98 7.00
2.2 × 10−6 3.27 3.27
99.99 53 53
Vu =
Vy =
T −1 1 X |u(k) − usp |2R + |u(k) − u(k − 1)|2S T
1 T
0 T −1 X
|y (k) − ysp |Q 2 y
0
V = Vu + Vy Rawlings/Angeli/Bates
Economic MPC
29 / 94
Example 2. Two input–two output system with noise
" G (s) =
1.5 (s+2)(s+1) 0.5 (s+0.5)(s+1)
0.75 (s+5)(s+2) 2 (s+2)(s+3)
#
Sample time T = 0.25 sec Input constraints −0.5 ≤ u1 , u2 ≤ 0.5 Setpoint ysp = [0.337 0.34]0 Qy = 5I , R = I , S = I Horizon length N = 80 Periodic disturbance d = ±[0.03 − 0.03]0 with Gd = G and measurement and state noise
Rawlings/Angeli/Bates
Economic MPC
30 / 94
0.338 0.337 0.336 0.335 y1 0.334 0.333 0.332
setpoint y1 (sp-MPC) y1 (targ-MPC)
0.331 0.33 0
5
10 15 Time (sec)
20
25
0.342 0.341 0.34 0.339 y2 0.338 0.337 setpoint y2 (sp-MPC) y2 (targ-MPC)
0.336 0.335 0 Rawlings/Angeli/Bates
5
10 15 Time (sec)
20
Economic MPC
25 31 / 94
0.5
u1 (targ-MPC) u1 (sp-MPC)
0.45
u1
0.5
u1 0.45 0
5
10 15 Time (sec)
20
25 -0.45 -0.465
u2
-0.48 u2 (targ-MPC) u2 (sp-MPC) -0.45 u2
-0.465 -0.48 0
Rawlings/Angeli/Bates
5
10 15 Time (sec)
20
Economic MPC
25 32 / 94
y1s 0.3 setpoint target (y1s ) db1
0.2 0.1 0
db1
-0.1 0
5
10 15 Time (sec)
20
25
y2s 0.3 setpoint target (y2s ) db2
0.2
0.1
db2
0 0
Rawlings/Angeli/Bates
5
10 15 Time (sec)
20
Economic MPC
25 33 / 94
Summary of Example 2
Performance Measure
targ-MPC (×10−4 )
sp-MPC (×10−4 )
∆(index)%
Vu Vy V
3.32 1.63 4.95
2.10 0.04 2.14
37 98 57
Rawlings/Angeli/Bates
Economic MPC
34 / 94
Economic MPC: motivating the idea
Profit
-4
Rawlings/Angeli/Bates
-2 0 Input (u)
2
4
Economic MPC
-4
-2
0
4 2 State (x)
35 / 94
Economic MPC: motivating the idea
Profit
-4
Rawlings/Angeli/Bates
-2 0 Input (u)
2
4
Economic MPC
-4
-2
0
4 2 State (x)
35 / 94
Economic MPC definition (with terminal constraint)
Economic stage cost: `(x, u)
Rawlings/Angeli/Bates
Economic MPC
36 / 94
Economic MPC definition (with terminal constraint)
Economic stage cost: `(x, u) Optimization: min VN,e (x, u) = u
N−1 X
`(x(k), u(k))
k=0
+ x = f (x, u) x(0) = x (x(k), u(k)) ∈ Z k ∈ [0 : N − 1] subject to x(N) = xs Control law: u = κN,e (x) Admissible set: XN,e
Rawlings/Angeli/Bates
Economic MPC
36 / 94
(1)
Example
x + = Ax + Bu 0.857 0.884 8.565 A= B= −0.0147 −0.0151 0.88418 Input constraint: −1 ≤ u ≤ 1 `(x, u) = α0 x + β 0 u 0 α = −3 −2 β = −2
Rawlings/Angeli/Bates
`t (x, u) = |x − xs |2Q + |u − us |2R Q = 2I2 R=2 0 xs = 60 0 us = 1
Economic MPC
37 / 94
10 8 6 x2
4 2 0 -2 60
Rawlings/Angeli/Bates
65 70 targ-MPC x1
75
Economic MPC
80
85
38 / 94
10 8 6 x2
4 2 0 -2 60
Rawlings/Angeli/Bates
65 70 targ-MPC x1
75
Economic MPC
80 85 eco-MPC
38 / 94
80
targ-MPC
State 1
75 70 65 60
State 2
55 10 8 6 4 2 0 -2
0
2
4
6
8
10
12
14
targ-MPC
0
2
4
6
8
10
12
14
1 Input
0.5 0 -0.5 -1 0
2
4
6
8
10
targ-MPC 12
14
Time
Rawlings/Angeli/Bates
Economic MPC
39 / 94
State 1
90 85 80 75 70 65 60 55
State 2
10 8 6 4 2 0 -2
targ-MPC eco-MPC
0
2
4
6
8
10
12
14
targ-MPC eco-MPC
0
2
4
6
8
0
2
4
6
8
10
12
14
1 Input
0.5 0 -0.5 -1 10
targ-MPC eco-MPC 12
14
Time
Rawlings/Angeli/Bates
Economic MPC
39 / 94
Closed-loop performance measures Profitability: I
Average asymptotic cost relative to steady state T 1 X `(x(k), u(k)) − `(xs , us ) T →∞ T
lim
k=0
I
Net cost relative to steady state ∞ X k=0
`(x(k), u(k)) − `(xs , us )
Stability: I
Asymptotic convergence to optimal steady state lim (x(k), u(k)) = (xs , us )
k→∞
Rawlings/Angeli/Bates
Economic MPC
40 / 94
How can profitability and stability be opposing goals? We consider a nonlinear constant volume isothermal CSTR I I
State: CA Input: CAf
The following reactions take place: r = kcA2
A→B Economic stage cost:
`(x, u) = −CB Input constraints over horizon of N: 0 ≤ u(k) ≤ 3
Rawlings/Angeli/Bates
N 1 X u(k) = 1 N k=0
Economic MPC
41 / 94
Optimal control solution
Optimal u and x 0.5
4
0.4 3 0.3
u
2
x 0.2
1 0.1
0
0 0
20
40
t
60
Rawlings/Angeli/Bates
80
100
Economic MPC
42 / 94
Optimal control solution
Production rate, RB = kcA2
Optimal u and x 4
0.5
0.3
0.4
0.25
cAs
3 0.2 0.3
u
2
cA2
x
0.15
0.2 0.1
hcA2 i
1 0.1 0.05 0
0 0
20
40
t
60
Rawlings/Angeli/Bates
80
100
2 cAs
0 0
Economic MPC
20
40
t
60
80
42 / 94
100
Average economic performance of EMPC
When considering economic optimization as the objective of control, average economic performance is the more natural performance measure. In tracking MPC, average closed-loop economic performance is guaranteed indirectly, via stability T 1 X `(x(k), u(k)) = `(xs , us ) lim (x(k), u(k)) = (xs , us ) ⇒ lim k→∞ T →∞ T k=0
Several methods are available for stabilizing tracking MPC including the addition of a terminal equality constraint, x(N) = xs , and a terminal penalty
Rawlings/Angeli/Bates
Economic MPC
43 / 94
What does a terminal constraint accomplish for EMPC?
Theorem: Average economic performance of EMPC The average asymptotic cost of the closed-loop system x + = f (x, κN,e (x)) satisfies
PT lim sup T →+∞
Rawlings/Angeli/Bates
k=0 `(x(k), u(k))
T +1
≤ `(xs , us )
Economic MPC
44 / 94
What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state
Rawlings/Angeli/Bates
Economic MPC
45 / 94
What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I
EMPC is stable under these conditions
Rawlings/Angeli/Bates
Economic MPC
45 / 94
What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
Rawlings/Angeli/Bates
Economic MPC
45 / 94
What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state? ∞ X k=0
Rawlings/Angeli/Bates
`(x(k), u(k)) − `(xs , us )
Economic MPC
45 / 94
What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state? ∞ X k=0 I
`(x(k), u(k)) − `(xs , us )
Bounded above because VN0 (x) is bounded on XN .
Rawlings/Angeli/Bates
Economic MPC
45 / 94
What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state? ∞ X k=0 I I
`(x(k), u(k)) − `(xs , us )
Bounded above because VN0 (x) is bounded on XN . Not bounded below because the controller can outperform the best steady state on average
Rawlings/Angeli/Bates
Economic MPC
45 / 94
What this means . . . The nominal average asymptotic cost of closed-loop EMPC is not worse than the best steady state What this theorem does not say: I I
EMPC is stable under these conditions A finite time average is not worse than the best steady state
What about net, rather than average, cost relative to steady state? ∞ X k=0 I I
I
`(x(k), u(k)) − `(xs , us )
Bounded above because VN0 (x) is bounded on XN . Not bounded below because the controller can outperform the best steady state on average No proven relationship between closed-loop cost for tracking MPC vs. EMPC from a given initial state
Rawlings/Angeli/Bates
Economic MPC
45 / 94
Industrial simulation example F200
F4 , T3
Separator L2
Cooling water T200
T201 Condenser
Condensate F5
P100 LT Steam F100
T100
Evaporator
Condensate
LC
F3
F2
Feed F1 , X1 , T1
Product X2 , T2
Evaporator system (Newell and Lee, 1989, Ch. 2) Rawlings/Angeli/Bates
Economic MPC
46 / 94
Evaporator system Measurements: product composition X2 and operating pressure P2 Inputs: steam pressure P100 , cooling water flow rate F200 The economic stage cost is the operating cost for electricity, steam and cooling water (Wang and Cameron, 1994; Govatsmark and Skogestad, 2001). J = 1.009(F2 + F3 ) + 600F100 + 0.6F200
Rawlings/Angeli/Bates
Economic MPC
47 / 94
Evaporator system Measurements: product composition X2 and operating pressure P2 Inputs: steam pressure P100 , cooling water flow rate F200 The economic stage cost is the operating cost for electricity, steam and cooling water (Wang and Cameron, 1994; Govatsmark and Skogestad, 2001). J = 1.009(F2 + F3 ) + 600F100 + 0.6F200
We consider the process subject to disturbances in feed flow rate F1 , Feed composition C1 , Circulating flow rate F3 , feed temperature T1 and cooling water inlet temperature T200
Rawlings/Angeli/Bates
Economic MPC
47 / 94
Evaporator system Measurements: product composition X2 and operating pressure P2 Inputs: steam pressure P100 , cooling water flow rate F200 The economic stage cost is the operating cost for electricity, steam and cooling water (Wang and Cameron, 1994; Govatsmark and Skogestad, 2001). J = 1.009(F2 + F3 ) + 600F100 + 0.6F200
We consider the process subject to disturbances in feed flow rate F1 , Feed composition C1 , Circulating flow rate F3 , feed temperature T1 and cooling water inlet temperature T200 We consider both tracking MPC and EMPC with a terminal state constraint Rawlings/Angeli/Bates
Economic MPC
47 / 94
14000 12000 Cost ($/h)
Disturbance P2 (kPa)
0 -1 -2 -3 -4 -5 -6 -7 -8 -9
0
20
40 60 time (min)
80
0
20
40 60 time (min)
80
100
0
20
40 60 time (min)
80
100
80
100
220 Input F200 (kg/min)
Input P100 (kPa)
6000 2000
100
350 300 250 200 150 100
0
20
40 60 time (min)
80
200 180 160 140 120 100
100
70
50
60
Output P2 (kPa)
Output X2(%)
8000 4000
400
50 40 30 20
10000
0
20
Rawlings/Angeli/Bates
40 60 time (min)
80
100
eco-MPC track-MPC
45 40 35 30
0
20
Economic MPC
40 60 time (min)
48 / 94
12000
F1 T1
Cost ($/h)
Disturbance
50 45 40 35 30 25 20 15 10
T100 0 10 20 30 40 50 60 70 80 90 time (min)
6000 0 10 20 30 40 50 60 70 80 90 time (min)
400 Input F200 (kg/min)
Input P100 (kPa)
8000
4000
400 350 300 250 200 150 100
10000
360 320 280 240 200
0 10 20 30 40 50 60 70 80 90 time (min)
0 10 20 30 40 50 60 70 80 90 time (min)
36
Output P2 (kPa)
Output X2(%)
40
32 28 24 20
0 10 20 30 40 50 60 70 80 90 time (min)
Rawlings/Angeli/Bates
56
eco-MPC track-MPC
54 52 50 0 10 20 30 40 50 60 70 80 90 time (min)
Economic MPC
49 / 94
Evaporator system closed-loop economics
Performance comparison under economic and tracking MPC
Disturbance Measured Unmeasured
Rawlings/Angeli/Bates
Avg. operating cost ×10−6 ($/ hr) eco-MPC track-MPC ∆ (%) 5.89 5.80
5.97 6.15
Economic MPC
2.2 6.2
50 / 94
Asymptotic stability and EMPC
Steady operation often desired by practitioners I I
Equipment not designed for strongly unsteady operation Operator acceptance issue for unsteady operation
Rawlings/Angeli/Bates
Economic MPC
51 / 94
Asymptotic stability and EMPC
Steady operation often desired by practitioners I I
Equipment not designed for strongly unsteady operation Operator acceptance issue for unsteady operation
Stability analysis I
I
Check that stability is consistent with the process model and control objectives Or modify the control objectives (stage cost) given the process model
Rawlings/Angeli/Bates
Economic MPC
51 / 94
Stabilizing assumption for EMPC System: x + = f (x, u) Supply rate: s(x, u)
Storage: λ(x)
Dissipation
Assumption: Dissipativity The system x + = f (x, u) is dissipative with respect to the supply rate s : Z → R if there exists a function λ : X → R such that: λ(f (x, u)) − λ(x) ≤ s(x, u) for all (x, u) ∈ Z. If ρ : X → R≥0 positive definite exists such that: λ(f (x, u)) − λ(x) ≤ −ρ(x) + s(x, u) then the system is said to be strictly dissipative. Rawlings/Angeli/Bates
Economic MPC
52 / 94
Optimality of steady-state operation Optimal operation at steady-state The system x + = f (x, u) is optimally operated at steady-state if for all trajectories (x, u), such that (x(k), u(k)) ∈ Z for all k ∈ N it holds that PT −1 lim inf
T →+∞
Rawlings/Angeli/Bates
k=0
`(x(k), u(k)) ≥ `(xs , us ) T
Economic MPC
53 / 94
Optimality of steady-state operation Optimal operation at steady-state The system x + = f (x, u) is optimally operated at steady-state if for all trajectories (x, u), such that (x(k), u(k)) ∈ Z for all k ∈ N it holds that PT −1 lim inf
T →+∞
k=0
`(x(k), u(k)) ≥ `(xs , us ) T
Suboptimal operation off steady-state A system optimally operated at steady-state is suboptimally operated off steady-state if for all trajectories (x, u), such that (x(k), u(k)) ∈ Z it holds that either PT −1 k=0 `(x(k), u(k)) lim inf > `(xs , us ) T →+∞ T or lim inf |x(k) − xs | = 0. k→+∞
Rawlings/Angeli/Bates
Economic MPC
53 / 94
Dissipativity and Optimality
Dissipativity is closely related to optimal operation at steady-state 1 2
Dissipativity ⇒ Optimal operation at steady-state
Strict Dissipativity ⇒ Sub-optimal operation off steady-state
Rawlings/Angeli/Bates
Economic MPC
54 / 94
Dissipativity and Optimality
Dissipativity is closely related to optimal operation at steady-state 1 2 3
Dissipativity ⇒ Optimal operation at steady-state
Strict Dissipativity ⇒ Sub-optimal operation off steady-state
Gap between Dissipativity and Optimal steady state operation: ∀ x, all trajectories (x, u) such that (x(k), u(k)) ∈ Z and x(0) = x satisfy inf
T ≥1
T −1 X k=0
`(x(k), u(k)) − `(xs , us ) > −∞
investigated by Mueller et al. (2013)
Rawlings/Angeli/Bates
Economic MPC
54 / 94
Dissipativity: an example
Consider the following system x + = αx + (1 − α)u,
α ∈ [0, 1)
With the following nonconvex stage cost `(x, u) = (x + u/3)(2u − x) + (x − u)4
1
Strong duality requires constant λ such that λ0 f (x, u) − λ0 x ≤ `(x, u). Rawlings/Angeli/Bates
Economic MPC
55 / 94
Dissipativity: an example
Consider the following system x + = αx + (1 − α)u,
α ∈ [0, 1)
With the following nonconvex stage cost `(x, u) = (x + u/3)(2u − x) + (x − u)4 This system and stage cost are dissipative for α ∈ [ 12 , 1], but are not strongly dual1 for any α. What does this tell us about the behavior of EMPC?
1
Strong duality requires constant λ such that λ0 f (x, u) − λ0 x ≤ `(x, u). Rawlings/Angeli/Bates
Economic MPC
55 / 94
0.2 0.15 0.1 State
0.05 0 -0.05 -0.1 -0.15 -0.2 0
2
4
6
α = 0.2
8
10 Time
12
α = 0.4
14
16
18
α = 0.6
0.3 0.2
Input
0.1 0 -0.1 -0.2 -0.3 0
2
α = 0.2 Rawlings/Angeli/Bates
4
6
8
10 Time
α = 0.4
12
14
16
18
α = 0.6 Economic MPC
56 / 94
0.05
α = .2 α = .4 α = .6
0.04
Stage Cost
0.03 0.02 0.01 0 -0.01 -0.02 -0.03 0
2
4
6
8
10 Time
12
14
16
18
EMPC controller with N = 10, terminal equality constraint Closed-loop EMPC is stable for α ≥ 1/2
Closed-loop EMPC is unstable for α < 1/2 and outperforms the best steady state on average Dissipativity is a tighter condition for EMPC stability than strong duality Rawlings/Angeli/Bates
Economic MPC
57 / 94
u
α u = x/2
(b) x
(a)
u = −3x
Qualitative picture of L0 : global minima (a),(b); α-dependent period-2 solution (black dots). Adapted from Angeli et al. (2012).
Rawlings/Angeli/Bates
Economic MPC
58 / 94
Stability theorem for EMPC
Theorem: Stability of EMPC If the system x + = f (x, κN,e (x)) is strictly dissipative with respect to the supply rate s(x, u) = `(x, u) − `(xs , us ) then xs is an asymptotically stable equilibrium point of the closed-loop system with region of attraction XN,e . Nominal average asymptotic performance not worse than steady operation is always implied by stability
Rawlings/Angeli/Bates
Economic MPC
59 / 94
Sketch of proof Lyapunov-based proof, with rotated stage cost: L(x, u) := `(x, u) − `(xs , us ) + λ(x) − λ(f (x, u))
Rawlings/Angeli/Bates
Economic MPC
60 / 94
Sketch of proof Lyapunov-based proof, with rotated stage cost: L(x, u) := `(x, u) − `(xs , us ) + λ(x) − λ(f (x, u)) Notice that N−1 X
L(x(k), u(k))
k=0
=
N−1 X k=0
`(x(k), u(k)) + λ(x(k)) − λ(x(k + 1)) − `(xs , us )
= λ(x(0)) − λ(x(N)) − N`(xs , us ) +
N−1 X
`(x(k), u(k))
k=0
Optimal control is unaffected by cost rotation. Rawlings/Angeli/Bates
Economic MPC
60 / 94
Sketch of proof (continued) Under dissipativity assumption rotated cost fulfills standard MPC conditions (positive semi-definite): L(x, u) ≥ 0 In addition, under strict dissipativity, the rotated cost is positive definite: L(x, u) ≥ ρ(x)
Rawlings/Angeli/Bates
Economic MPC
61 / 94
Sketch of proof (continued) Under dissipativity assumption rotated cost fulfills standard MPC conditions (positive semi-definite): L(x, u) ≥ 0 In addition, under strict dissipativity, the rotated cost is positive definite: L(x, u) ≥ ρ(x) Cost-to-go can be used as a candidate Lyapunov function V (x) := min u
N−1 X
L(x(k), u(k))
k=0
subject to initial, terminal and dynamic constraints. Rawlings/Angeli/Bates
Economic MPC
61 / 94
Sketch of proof (continued) Recursive feasibility: Feasibility of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: x = (x(1), x(2), . . . , x(N − 1), xs ) u = (u(1), . . . , u(N − 1), us ).
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Sketch of proof (continued) Recursive feasibility: Feasibility of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: x = (x(1), x(2), . . . , x(N − 1), xs ) u = (u(1), . . . , u(N − 1), us ). Along closed-loop solutions: V (x + ) ≤ V (x) − ρ(x). Hence, ρ(x(k)) → 0 as k → +∞ and convergence to equilibrium follows by compactness of Z. Rawlings/Angeli/Bates
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Asymptotic averages As convergence to equilibrium is not always enforced, asymptotic averages of output variables need not equal their limit (which in fact may fail to exist) Possibility of asymptotic average constraints different from pointwise in time constraints (pointwise in time constraints are typically more stringent)
Rawlings/Angeli/Bates
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Asymptotic averages As convergence to equilibrium is not always enforced, asymptotic averages of output variables need not equal their limit (which in fact may fail to exist) Possibility of asymptotic average constraints different from pointwise in time constraints (pointwise in time constraints are typically more stringent) Definition of asymptotic average: Av [v ] = {w : ∃ {Tn }+∞ n=1 : Tn → +∞ as n → +∞ PTn −1 k=0 v (k) and lim =w } n→+∞ Tn Asymptotic average need not be a singleton: 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, . . . Notice that {1/3, 1/2} ⊂ Av[v ] ⊂ [1/3, 1/2]. Rawlings/Angeli/Bates
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Average constraints Goals: Modify economic MPC algorithm to guarantee: (x(k), u(k)) ∈ Z for all k ∈ N; Av[h(x, u)] ⊂ Y 3 h(xs , us ); Recursive feasibility
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Average constraints Goals: Modify economic MPC algorithm to guarantee: (x(k), u(k)) ∈ Z for all k ∈ N; Av[h(x, u)] ⊂ Y 3 h(xs , us ); Recursive feasibility Remarks: 1
For technical reasons Y is assumed to be convex.
2
Average constraint does not imply limits on averages computed on finite time windows.
For the following: At each time t let variables z(k) and v (k) denote virtual (predicted) variables, and x(k), u(k) (k ≤ t) actual variables Rawlings/Angeli/Bates
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Economic MPC with average constraints A modified receding-horizon strategy: min v,z
N−1 X
`(z(k), v (k))
k=0
subject to: (z(k), v (k)) ∈ Z ∀ k ∈ {0, 1, . . . , N − 1}
z(k + 1) = f (z(k), v (k)) ∀ k ∈ {0, 1, . . . , N − 1} z(0) = x(t) N−1 X k=0
z(N) = xs
h(z(k), v (k)) ∈ Yt
where: Yt+1 = Yt ⊕ Y {h(x(t), u(t))}
Y0 = NY ⊕ Y00
with ⊕, denoting set addition and subtraction, and Y00 is an arbitrary convex compact set. Rawlings/Angeli/Bates
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Properties of the algorithm Time-varying state-feedback Recursive feasibility follows by standard argument: Feasibility of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: x = (x(1), x(2), . . . , x(N − 1), xs ) u = (u(1), . . . , u(N − 1), us ).
Rawlings/Angeli/Bates
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Properties of the algorithm Time-varying state-feedback Recursive feasibility follows by standard argument: Feasibility of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: x = (x(1), x(2), . . . , x(N − 1), xs ) u = (u(1), . . . , u(N − 1), us ). Additional constraint does not limit feasibility region (Y0 is arbitrarily large) Asymptotic average constraints are guaranteed Average performance not worse than best steady-state Possibility of replacing terminal constraint with terminal penalty function Rawlings/Angeli/Bates
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Working principle Show by induction: Yt = tY + Y0 {
t−1 X
h(x(k), u(k))}
k=0
Hence: N−1 X
h(z(k), v (k)) +
k=0
Rawlings/Angeli/Bates
t−1 X k=0
h(x(k), u(k)) ∈ tY ⊕ Y0
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Working principle Show by induction: Yt = tY + Y0 {
t−1 X
h(x(k), u(k))}
k=0
Hence: N−1 X
h(z(k), v (k)) +
k=0
t−1 X k=0
h(x(k), u(k)) ∈ tY ⊕ Y0
Taking tn → +∞ so that limit exists: Ptn −1 k=0 h(x(k), u(k)) = lim n→+∞ tn PN−1 Ptn −1 k=0 h(z(k), v (k)) + k=0 h(x(k), u(k)) = lim ∈Y n→+∞ tn Rawlings/Angeli/Bates
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Dissipativity for averagely constrained systems
Problem: Asymptotic convergence of averagely constrained MPC Sufficient conditions, possibility of relaxing dissipativity?
Dissipativity taking into account average constraints Consider the modified supply function: ¯ T [Ah(x, u) − b] sa (x, u) = `(x, u) − `(xs , us ) + λ provided: Y = {y : Ay ≤ b}
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Dissipativity and Convergence for Averagely Constrained EMPC Optimal steady-state operation If a system is dissipative with respect to the supply rate sa (x, u) for some ¯ then every feasible solution fulfills: non-negative λ lim inf
T →+∞
Rawlings/Angeli/Bates
T −1 X k=0
`(x(k), u(k)) ≥ `(xs , us ) T
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Dissipativity and Convergence for Averagely Constrained EMPC Optimal steady-state operation If a system is dissipative with respect to the supply rate sa (x, u) for some ¯ then every feasible solution fulfills: non-negative λ lim inf
T →+∞
T −1 X k=0
`(x(k), u(k)) ≥ `(xs , us ) T
Convergence of Averagely Constrained EMPC If a system is strictly dissipative with respect to the supply rate sa (x, u) for ¯ then closed-loop solutions of Averagely Constrained some non-negative λ EMPC asymptotically approach the best steady-state.
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Enforcing convergence in Economic MPC
By modifying the economic stage cost `(x, u) as: ˜ u) = `(x, u) + γ(|x − xs | + |u − us |) `(x, in order to recover dissipativity or strong duality
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Enforcing convergence in Economic MPC
By modifying the economic stage cost `(x, u) as: ˜ u) = `(x, u) + γ(|x − xs | + |u − us |) `(x, in order to recover dissipativity or strong duality By adding an auxiliary average constraint, such as: Av[|x − xs |2 ] ⊂ {0} or: Av[|xi − xsi |2 ] ⊂ {0}
Rawlings/Angeli/Bates
i = 1 . . . n;
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Case study: a reactor with parallel reactions The reactions: R → P1
R → P2
The state-space model: x˙ 1 = 1 − 104 x12 e −1/x3 − 400x1 e −0.55/x3 − x1 x˙ 2 = 104 x12 e −1/x3 − x2 x˙ 3 = u − x3
x1 is concentration of R, x2 is concentration of P1 (the desired product), x3 is the temperature and u is the heat flux. P2 is the waste product (not modeled).
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Case study: a reactor with parallel reactions The reactions: R → P1
R → P2
The state-space model: x˙ 1 = 1 − 104 x12 e −1/x3 − 400x1 e −0.55/x3 − x1 x˙ 2 = 104 x12 e −1/x3 − x2 x˙ 3 = u − x3
x1 is concentration of R, x2 is concentration of P1 (the desired product), x3 is the temperature and u is the heat flux. P2 is the waste product (not modeled). Goal: maximize x2 , viz.: `(x, u) = −x2 ; Known that best performance is not at steady state Rawlings/Angeli/Bates
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Closed-loop simulations
0.4 0.3 u 0.2 0.1 0
2
4
6
8
10
0.7 0.6 0.5 0.4 x1 0.3 0.2 0.1 00
0.2
IC 1 IC 2 IC 3
2
4
6
8
10
2
4
6
8
10
0.2 0.15
x2
x3 0.1 00
2
4
6
8
10
0.05 0
Closed-loop input and state profiles for economic MPC with a convex term and different initial states
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Closed-loop simulations (2)
0.4 0.3 u 0.2 0.1 0
2
4
6
8
10
0.7 0.6 0.5 0.4 x1 0.3 0.2 0.1 00
0.2
IC 1 IC 2 IC 3
2
4
6
8
10
2
4
6
8
10
0.2 0.15
x2
x3 0.1 00
2
4
6
8
10
0.05 0
Closed-loop input and state profiles for economic MPC with a convergence constraint and different initial states
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Outperforming best steady-state
Terminal constraint instrumental in: 1 2
guaranteeing recursive feasibility providing bound to asymptotic performance
Any feasible trajectory may be used as a terminal constraint
Idea: Replace best equilibrium by best feasible periodic solution of given period
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Best feasible periodic solution Let xs , us be: xs = [xs (0), xs (1), . . . , xs (Q − 1)] us = [us (0), us (1), . . . , us (Q − 1)], and assume that they belong to: arg min x,u
Q−1 X
`(x(k), u(k))
k=0
subject to: x(k + 1) = f (x(k), u(k))
k ∈ {0, 1, . . . , Q − 2}
x(0) = f (x(Q − 1), u(Q − 1)) (x(k), u(k)) ∈ Z
k ∈ {0, 1, . . . , Q − 1}
We call xs , us an optimal Q-periodic solution. Rawlings/Angeli/Bates
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EMPC with periodic terminal constraint
Solve at each time t the following optimization problem: min v,z
N−1 X
`(z(k), v (k))
k=0
subject to: (z(k), v (k)) ∈ Z ∀ k ∈ {0, 1, . . . , N − 1} z(k + 1) = f (z(k), v (k)) ∀ k ∈ {0, 1, . . . , N − 1} z(0) = x(t)
Rawlings/Angeli/Bates
z(N) = xs (t mod Q)
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Features of the algorithm
Q periodic state feedback Recursive feasibility: feasibility at time t of: x = (x(0), x(1), . . . , x(N − 1))
u = (u(0), u(1), . . . , u(N − 1))
implies feasibility of: (x(1), . . . , x(N−1), xs (t mod Q))
(u(1), . . . , u(N−1), us (t mod Q))
at time t + 1. Possibility of incorporating average constraints
Rawlings/Angeli/Bates
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Average performance with periodic end constraint Let V be the cost to go: V (t, x) = min v,z
N−1 X
`(z(k), v (k))
k=0
subject to: (z(k), v (k)) ∈ Z ∀ k ∈ {0, 1, . . . , N − 1} z(k + 1) = f (z(k), v (k)) ∀ k ∈ {0, 1, . . . , N − 1} z(0) = x
Rawlings/Angeli/Bates
z(N) = xs (t mod Q)
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Average performance with periodic end constraint Let V be the cost to go: V (t, x) = min v,z
N−1 X
`(z(k), v (k))
k=0
subject to: (z(k), v (k)) ∈ Z ∀ k ∈ {0, 1, . . . , N − 1} z(k + 1) = f (z(k), v (k)) ∀ k ∈ {0, 1, . . . , N − 1} z(0) = x
z(N) = xs (t mod Q)
Along closed-loop solution: V (t + 1, x(t + 1)) ≤ V (t, x(t))
− `(x(t), u(t)) + `(xs (t mod Q), us (t mod Q))
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Average performance with periodic end constraint
Taking sums between 0 and T − 1 and dividing by T yields: PT −1 lim sup T →+∞
k=0
`(x(k), u(k)) ≤ T
PQ−1 k=0
`(xs (k), us (k)) Q
Average performance at least as good as optimal Q-periodic solution
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Average performance with periodic end constraint
Taking sums between 0 and T − 1 and dividing by T yields: PT −1 lim sup T →+∞
k=0
`(x(k), u(k)) ≤ T
PQ−1 k=0
`(xs (k), us (k)) Q
Average performance at least as good as optimal Q-periodic solution Q and N may be different from each other and unrelated The closed-loop system need not be asymptotically stable to the optimal Q periodic solution The optimal Q periodic solution need not be an equilibrium of the closed-loop system
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Ammonia synthesis (Jain et al., 1985)
Open loop stable equilibria Open-loop periodic forcing improves average production rate 1 2
Improvement of 24% by using sinusoidal inputs u(t) = c + A sin(ωt) Improvement of 25% by using square-waves of different amplitudes, average and duty cycle
Rawlings/Angeli/Bates
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Simulation of EMPC with terminal steady-state constraint
Improvement of 30 % in Ammonia production rate
Rawlings/Angeli/Bates
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Best Q-periodic solutions
Discretization Ts = 0.1 Best Q-periodic solution for Q = 16 Average production rate: 50 % better than best square wave Rawlings/Angeli/Bates
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Closed-loop with Q periodic terminal constraint
Chaotic regime: 25 % better than periodic solution Overall ≈ 140 % improvement over best steady state Rawlings/Angeli/Bates
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Terminal penalty formulations Just as with tracking MPC, we can expand the feasible set XN by replacing the terminal equality constraint with a terminal set constraint and a terminal penalty Admissible set: XN,p ={x ∈ X | ∃ u such that the trajectory (x(k), u(k)) satisfies (x(k), u(k)) ∈ Z k ∈ I0:N−1 ,
x(N) ∈ Xf }
Objective function: VN,p (x, u) =
N−1 X
`(x(k), u(k)) + Vf (x(N))
k=0
Rawlings/Angeli/Bates
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Terminal penalty in EMPC
Assumption: Terminal penalty There exists a terminal region control law κf : Xf → U such that Vf (f (x, κf (x))) ≤ Vf (x) − `(x, κf (x)) + `(xs , us ) (x, κf (x)) ∈ ZN,p
∀x ∈ Xf
This assumption is identical in form to the tracking case, but `(x, u) has changed Vf (x) need not be positive definite with respect to xs
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Terminal penalty in EMPC
Theorem: EMPC stability with terminal penalty If the system x + = f (x, κN,p (x)) is strictly dissipative with respect to the supply rate: s(x, u) = `(x, u) − `(xs , us ) then xs is an asymptotically stable equilibrium point of the closed-loop system with region of attraction XN .
Rawlings/Angeli/Bates
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Case study: nonlinear CSTR We consider a nonlinear constant volume isothermal CSTR I I
States: P0 , B, P1 , P2 Inputs: inflow concentrations of P0 , B
The following reactions take place: P0 + B −→ P1 P1 + B −→ P2
Rawlings/Angeli/Bates
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Case study: nonlinear CSTR We consider a nonlinear constant volume isothermal CSTR I I
States: P0 , B, P1 , P2 Inputs: inflow concentrations of P0 , B
The following reactions take place: P0 + B −→ P1 P1 + B −→ P2 Economic stage cost: `(x, u) = −CP1 The controller stage cost is modified according to: ˜ u) = `(x, u) + |x − xs |2 + |u − us |2 `(x, Q R Rawlings/Angeli/Bates
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Representative open-loop trajectories 2
6
1.5
4
x1 1
x2 2
0.5 00
5
10
15
1.5
00 0.8
1 x4 0.4
x3
5
10
15
5 10 Time (t)
15
Eco R =0 Str. dual track
0.5 00
5 10 Time (t)
15
00
Open-loop state profiles with different cost functions and arbitrary initial state.
Rawlings/Angeli/Bates
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Average and net profits
Case Unstable Stable
Economic Dissipative Strongly dual Tracking
avg profit
net profit
0.46 0.38 0.38 0.38
∞ 2.6 1.6 1.0
Average profit and net profit for open-loop system with different stage costs.
All stable schemes have about the same average profit (large N) Net profit decreases with increase in tracking term weight
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Conclusions
The economic objective function of EMPC causes novel behavior I
EMPC may be unstable where MPC is stable
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Conclusions
The economic objective function of EMPC causes novel behavior I I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Rawlings/Angeli/Bates
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC
Rawlings/Angeli/Bates
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC I
Terminal penalty formulation
Rawlings/Angeli/Bates
Economic MPC
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC I I
Terminal penalty formulation Average constraints
Rawlings/Angeli/Bates
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Conclusions
The economic objective function of EMPC causes novel behavior I I
I
EMPC may be unstable where MPC is stable Using a terminal penalty or terminal equality constraint guarantees asymptotic average profit not worse than best steady state Stability of EMPC requires dissipativity of process/stage cost
Many techniques from MPC can be applied to EMPC I I I
Terminal penalty formulation Average constraints Periodic terminal constraints
Rawlings/Angeli/Bates
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction.
Rawlings/Angeli/Bates
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009)
Rawlings/Angeli/Bates
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009) Analyzing closed-loop performance
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009) Analyzing closed-loop performance I
What can be proven about net closed-loop performance of tracking MPC relative to EMPC?
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009) Analyzing closed-loop performance I
I
What can be proven about net closed-loop performance of tracking MPC relative to EMPC? What is observed about differences in net closed-loop performance in simulations?
Rawlings/Angeli/Bates
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Open research issues
Remove terminal constraints and costs by choosing sufficient N. See Gr¨ une (2012a,b) for recent results in this direction. Generalized terminal state constraint. Terminate on the steady-state manifold and move the end location dynamically to the best steady state (Fagiano and Teel, 2012; Ferramosca et al., 2009) Analyzing closed-loop performance I
I
I
What can be proven about net closed-loop performance of tracking MPC relative to EMPC? What is observed about differences in net closed-loop performance in simulations? What model, stage cost and disturbance characteristics cause large performance differences between MPC and EMPC?
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Open research issues
Tuning EMPC and robustness I
For nondissipative process/stage costs, how should the stage cost be modified?
Rawlings/Angeli/Bates
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Open research issues
Tuning EMPC and robustness I
I
For nondissipative process/stage costs, how should the stage cost be modified? How robust is EMPC to model errors and disturbances?
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Open research issues
Tuning EMPC and robustness I
I I
For nondissipative process/stage costs, how should the stage cost be modified? How robust is EMPC to model errors and disturbances? How can economic risk be incorporated into the controller?
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Open research issues
Tuning EMPC and robustness I
I I
For nondissipative process/stage costs, how should the stage cost be modified? How robust is EMPC to model errors and disturbances? How can economic risk be incorporated into the controller?
Computational methods for implementing EMPC; strategies for adapting existing control hierarchies
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Further reading I D. Angeli, R. Amrit, and J. B. Rawlings. On average performance and stability of economic model predictive control. IEEE Trans. Auto. Cont., 57(7):1615–1626, 2012. R. Dorfman, P. Samuelson, and R. Solow. Linear Programming and Economic Analysis. McGraw-Hill, New York, 1958. L. Fagiano and A. R. Teel. Model predictive control with generalized terminal state constraint. In IFAC Conference on Nonlinear Model Predictive Control 2012, pages 299–304, Noordwijkerhout, the Netherlands, August 2012. A. Ferramosca, D. Limon, I. Alvarado, T. Alamo, and E. Camacho. MPC for tracking of constrained nonlinear systems. In IEEE Conference on Decision and Control (CDC), pages 7978–7983, Shanghai, China, 2009. M. S. Govatsmark and S. Skogestad. Control structure selection for an evaporation process. Comput. Chem. Eng., 9:657–662, 2001. L. Gr¨ une. NMPC without terminal constraints. In IFAC Conference on Nonlinear Model Predictive Control 2012, pages 1–13, Noordwijkerhout, the Netherlands, August 2012a. L. Gr¨ une. Economic receding horizon control without terminal constraints. Automatica, 2012b. Accepted for publication. Rawlings/Angeli/Bates
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Further reading II
A. K. Jain, R. R. Hudgins, and P. L. Silveston. Effectiveness factor under cyclic operation of a reactor. Can. J. Chem., 63:166–169, February 1985. M. Mueller, D. Angeli, and F. Allg¨ ower. On convergence of averagely constrained economic MPC and necessity of dissipativity for optimal steady-state operation. In American Control Conference, 2013. R. B. Newell and P. L. Lee. Applied Process Control – A Case Study. Prentice Hall, Sydney, 1989. J. B. Rawlings, D. Bonn´e, J. B. Jørgensen, A. N. Venkat, and S. B. Jørgensen. Unreachable setpoints in model predictive control. IEEE Trans. Auto. Cont., 53(9): 2209–2215, October 2008. F. Wang and I. Cameron. Control studies on a model evaporation process–constrained state driving with conventional and higher relative degree systems. J. Proc. Cont., 4 (2):59–75, 1994.
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